làm giúp mình với

Cho $a,b$ lá các số thực dương. chứng minh:$\sqrt[3]{\frac{a^3+b^3}{2}}\leq \frac{a^2+b^2}{a+b}$

Bất đẳng thức

làm giúp mình với

Cho $a,b$ là các số thực dương. chứng minh:$\sqrt[3]{\frac{a^3+b^3}{2}}\leq \frac{a^2+b^2}{a+b}$

Bất đẳng thức

Theo BĐT AM-GM cho 2 số thực dương ta cóbc(b2+c2)" role="presentation" style="font-size: 12.8px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">bc(b2+c2)−−−−−−−−−√bc(b2+c2) = 12" role="presentation" style="font-size: 12.8px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">12√12.2bc(b2+c2)" role="presentation" style="font-size: 12.8px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">2bc(b2+c2)−−−−−−−−−−√2bc(b2+c2)≤" role="presentation" style="font-size: 12.8px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">≤≤ 2bc22" role="presentation" style="font-size: 12.8px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">2bc22√2bc22+b222" role="presentation" style="font-size: 12.8px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">b222√b222+c222" role="presentation" style="font-size: 12.8px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">c222√c222 =(b+c)222" role="presentation" style="font-size: 12.8px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">(b+c)222√(b+c)222⇒a(b+c)bc(b2+c2)≥22a(b+c)(b+c)2=22ab+c" role="presentation" style="font-size: 16px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">⇒a(b+c)bc(b2+c2)−−−−−−−−−√≥22√a(b+c)(b+c)2=22√ab+c⇒a(b+c)bc(b2+c2)≥22a(b+c)(b+c)2=22ab+c tương tự:b(c+a)ca(c2+a2)≥22bc+a" role="presentation" style="font-size: 16px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">b(c+a)ca(c2+a2)−−−−−−−−−√≥22√bc+ab(c+a)ca(c2+a2)≥22bc+a c(a+b)ab(a2+b2)≥22ca+b" role="presentation" style="font-size: 16px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">c(a+b)ab(a2+b2)−−−−−−−−−√≥22√ca+bc(a+b)ab(a2+b2)≥22ca+bcộng cả 3 vế lại ta có:a(b+c)bc(b2+c2)+b(c+a)ca(c2+a2)≥22bc+a+c(a+b)ab(a2+b2)≥22ca+b≥22(ab+c+bc+a+ca+b)≥22.32=32" role="presentation" style="font-size: 16px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">a(b+c)bc(b2+c2)−−−−−−−−−√+b(c+a)ca(c2+a2)−−−−−−−−−√≥22√bc+a+c(a+b)ab(a2+b2)−−−−−−−−−√≥22√ca+b≥22√(ab+c+bc+a+ca+b)≥22√.32=32√a(b+c)bc(b2+c2)+b(c+a)ca(c2+a2)≥22bc+a+c(a+b)ab(a2+b2)≥22ca+b≥22(ab+c+bc+a+ca+b)≥22.32=32Tagsb(c+a)ca(c2+a2)≥22bc+a c(a+b)ab(a2+b2)≥22ca+b" role="presentation" style="font-size: 16px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">c(a+b)ab(a2+b2)−−−−−−−−−√≥22√ca+bc(a+b)ab(a2+b2)≥22ca+bcộng cả 3 vế lại ta có: 22√ca+b22√ca+ba(b+c)bc(b2+c2)+b(c+a)ca(c2+a2)≥22bc+a+c(a+b)ab(a2+b2)≥22ca+b≥22(ab+c+bc+a+ca+b)≥22.32=32" role="presentation" style="font-size: 16px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">a(b+c)bc(b2+c2)−−−−−−−−−√+b(c+a)ca(c2+a2)−−−−−−−−−√≥22√bc+a+c(a+b)ab(a2+b2)−−−−−−−−−√≥22√ca+b≥22√(ab+c+bc+a+ca+b)≥22√.32=32

Theo BĐT AM-GM cho 2 số thực dương ta cóbc(b2+c2)" role="presentation" style="font-size: 12.8px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">bc(b2+c2)−−−−−−−−−√bc(b2+c2) = 12" role="presentation" style="font-size: 12.8px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">12√12.2bc(b2+c2)" role="presentation" style="font-size: 12.8px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">2bc(b2+c2)−−−−−−−−−−√2bc(b2+c2)≤" role="presentation" style="font-size: 12.8px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">≤≤ 2bc22" role="presentation" style="font-size: 12.8px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">2bc22√2bc22+b222" role="presentation" style="font-size: 12.8px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">b222√b222+c222" role="presentation" style="font-size: 12.8px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">c222√c222 =(b+c)222" role="presentation" style="font-size: 12.8px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">(b+c)222√(b+c)222⇒a(b+c)bc(b2+c2)≥22a(b+c)(b+c)2=22ab+c" role="presentation" style="font-size: 16px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">⇒a(b+c)bc(b2+c2)−−−−−−−−−√≥22√a(b+c)(b+c)2=22√ab+c⇒a(b+c)bc(b2+c2)≥22a(b+c)(b+c)2=22ab+ctương tự:b(c+a)ca(c2+a2)≥22bc+a" role="presentation" style="font-size: 16px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">b(c+a)ca(c2+a2)−−−−−−−−−√≥22√bc+ab(c+a)ca(c2+a2)≥22bc+a c(a+b)ab(a2+b2)≥22ca+b" role="presentation" style="font-size: 16px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">c(a+b)ab(a2+b2)−−−−−−−−−√≥22√ca+bc(a+b)ab(a2+b2)≥22ca+bcộng cả 3 vế lại ta có:P≥22(ab+c+bc+a+ca+b)≥22.32=32" role="presentation" style="font-size: 16px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">P≥22√(ab+c+bc+a+ca+b)≥22√.32=32√P≥22(ab+c+bc+a+ca+b)≥22.32=32Tagsa(b+c)bc(b2+c2)−−−−−−−−−√+b(c+a)ca(c2+a2)−−−−−−−−−√≥22√bc+a+c(a+b)ab(a2+b2)−−−−−−−−−√≥22√ca+b≥22√(ab+c+bc+a+ca+b)≥22√.32=32

Theo BĐT AM-GM cho 2 số thực dương ta cóbc(b2+c2)" role="presentation" style="font-size: 12.8px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">bc(b2+c2)−−−−−−−−−√bc(b2+c2) = 12" role="presentation" style="font-size: 12.8px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">12√12.2bc(b2+c2)" role="presentation" style="font-size: 12.8px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">2bc(b2+c2)−−−−−−−−−−√2bc(b2+c2)≤" role="presentation" style="font-size: 12.8px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">≤≤ 2bc22" role="presentation" style="font-size: 12.8px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">2bc22√2bc22+b222" role="presentation" style="font-size: 12.8px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">b222√b222+c222" role="presentation" style="font-size: 12.8px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">c222√c222 =(b+c)222" role="presentation" style="font-size: 12.8px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">(b+c)222√(b+c)222⇒a(b+c)bc(b2+c2)≥22a(b+c)(b+c)2=22ab+c" role="presentation" style="font-size: 16px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">⇒a(b+c)bc(b2+c2)−−−−−−−−−√≥22√a(b+c)(b+c)2=22√ab+c⇒a(b+c)bc(b2+c2)≥22a(b+c)(b+c)2=22ab+ctương tự:b(c+a)ca(c2+a2)≥22bc+a" role="presentation" style="font-size: 16px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">b(c+a)ca(c2+a2)−−−−−−−−−√≥22√bc+ab(c+a)ca(c2+a2)≥22bc+a c(a+b)ab(a2+b2)≥22ca+b" role="presentation" style="font-size: 16px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">c(a+b)ab(a2+b2)−−−−−−−−−√≥22√ca+bc(a+b)ab(a2+b2)≥22ca+bcộng cả 3 vế lại ta có: 22√ca+b22√ca+ba(b+c)bc(b2+c2)+b(c+a)ca(c2+a2)≥22bc+a+c(a+b)ab(a2+b2)≥22ca+b≥22(ab+c+bc+a+ca+b)≥22.32=32" role="presentation" style="font-size: 16px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">a(b+c)bc(b2+c2)−−−−−−−−−√+b(c+a)ca(c2+a2)−−−−−−−−−√≥22√bc+a+c(a+b)ab(a2+b2)−−−−−−−−−√≥22√ca+b≥22√(ab+c+bc+a+ca+b)≥22√.32=32

Theo BĐT AM-GM cho 2 số thực dương ta cóbc(b2+c2)" role="presentation" style="font-size: 12.8px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">bc(b2+c2)−−−−−−−−−√bc(b2+c2) = 12" role="presentation" style="font-size: 12.8px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">12√12.2bc(b2+c2)" role="presentation" style="font-size: 12.8px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">2bc(b2+c2)−−−−−−−−−−√2bc(b2+c2)≤" role="presentation" style="font-size: 12.8px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">≤≤ 2bc22" role="presentation" style="font-size: 12.8px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">2bc22√2bc22+b222" role="presentation" style="font-size: 12.8px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">b222√b222+c222" role="presentation" style="font-size: 12.8px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">c222√c222 =(b+c)222" role="presentation" style="font-size: 12.8px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">(b+c)222√(b+c)222⇒a(b+c)bc(b2+c2)≥22a(b+c)(b+c)2=22ab+c" role="presentation" style="font-size: 16px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">⇒a(b+c)bc(b2+c2)−−−−−−−−−√≥22√a(b+c)(b+c)2=22√ab+c⇒a(b+c)bc(b2+c2)≥22a(b+c)(b+c)2=22ab+c tương tự:b(c+a)ca(c2+a2)≥22bc+a" role="presentation" style="font-size: 16px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">b(c+a)ca(c2+a2)−−−−−−−−−√≥22√bc+ab(c+a)ca(c2+a2)≥22bc+a c(a+b)ab(a2+b2)≥22ca+b" role="presentation" style="font-size: 16px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">c(a+b)ab(a2+b2)−−−−−−−−−√≥22√ca+bc(a+b)ab(a2+b2)≥22ca+bcộng cả 3 vế lại ta có:a(b+c)bc(b2+c2)+b(c+a)ca(c2+a2)≥22bc+a+c(a+b)ab(a2+b2)≥22ca+b≥22(ab+c+bc+a+ca+b)≥22.32=32" role="presentation" style="font-size: 16px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">a(b+c)bc(b2+c2)−−−−−−−−−√+b(c+a)ca(c2+a2)−−−−−−−−−√≥22√bc+a+c(a+b)ab(a2+b2)−−−−−−−−−√≥22√ca+b≥22√(ab+c+bc+a+ca+b)≥22√.32=32√a(b+c)bc(b2+c2)+b(c+a)ca(c2+a2)≥22bc+a+c(a+b)ab(a2+b2)≥22ca+b≥22(ab+c+bc+a+ca+b)≥22.32=32Tagsb(c+a)ca(c2+a2)≥22bc+a c(a+b)ab(a2+b2)≥22ca+b" role="presentation" style="font-size: 16px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">c(a+b)ab(a2+b2)−−−−−−−−−√≥22√ca+bc(a+b)ab(a2+b2)≥22ca+bcộng cả 3 vế lại ta có: 22√ca+b22√ca+ba(b+c)bc(b2+c2)+b(c+a)ca(c2+a2)≥22bc+a+c(a+b)ab(a2+b2)≥22ca+b≥22(ab+c+bc+a+ca+b)≥22.32=32" role="presentation" style="font-size: 16px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">a(b+c)bc(b2+c2)−−−−−−−−−√+b(c+a)ca(c2+a2)−−−−−−−−−√≥22√bc+a+c(a+b)ab(a2+b2)−−−−−−−−−√≥22√ca+b≥22√(ab+c+bc+a+ca+b)≥22√.32=32

Theo BĐT AM-GM cho 2 số thực dương ta cóbc(b2+c2)" role="presentation" style="font-size: 12.8px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">bc(b2+c2)−−−−−−−−−√bc(b2+c2) = 12" role="presentation" style="font-size: 12.8px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">12√12.2bc(b2+c2)" role="presentation" style="font-size: 12.8px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">2bc(b2+c2)−−−−−−−−−−√2bc(b2+c2)≤" role="presentation" style="font-size: 12.8px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">≤≤ 2bc22" role="presentation" style="font-size: 12.8px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">2bc22√2bc22+b222" role="presentation" style="font-size: 12.8px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">b222√b222+c222" role="presentation" style="font-size: 12.8px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">c222√c222 =(b+c)222" role="presentation" style="font-size: 12.8px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">(b+c)222√(b+c)222⇒a(b+c)bc(b2+c2)≥22a(b+c)(b+c)2=22ab+c" role="presentation" style="font-size: 16px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">⇒a(b+c)bc(b2+c2)−−−−−−−−−√≥22√a(b+c)(b+c)2=22√ab+c⇒a(b+c)bc(b2+c2)≥22a(b+c)(b+c)2=22ab+ctương tự:b(c+a)ca(c2+a2)≥22bc+a" role="presentation" style="font-size: 16px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">b(c+a)ca(c2+a2)−−−−−−−−−√≥22√bc+ab(c+a)ca(c2+a2)≥22bc+a c(a+b)ab(a2+b2)≥22ca+b" role="presentation" style="font-size: 16px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">c(a+b)ab(a2+b2)−−−−−−−−−√≥22√ca+bc(a+b)ab(a2+b2)≥22ca+bcộng cả 3 vế lại ta có:cộng cả 3 vế lại ta có:a(b+c)bc(b2+c2)+b(c+a)ca(c2+a2)≥22bc+a+c(a+b)ab(a2+b2)≥22ca+b≥22(ab+c+bc+a+ca+b)≥22.32=32" role="presentation" style="font-size: 12.8px; position: relative;">a(b+c)bc(b2+c2)−−−−−−−−−√+b(c+a)ca(c2+a2)−−−−−−−−−√≥22√bc+a+c(a+b)ab(a2+b2)−−−−−−−−−√≥22√ca+b≥22√(ab+c+bc+a+ca+b)≥22√.32=32bc(b2+c2)−−−−−−−−−√+b(c+a)ca(c2+a2)−−−−−−−−−√≥22√bc+a+c(a+b)ab(a2+b2)−−−−−−−−−√≥22√ca+b≥22√(ab+c+bc+a+ca+b)≥22√.32=32√

Theo BĐT AM-GM cho 2 số thực dương ta cóbc(b2+c2)" role="presentation" style="font-size: 12.8px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">bc(b2+c2)−−−−−−−−−√bc(b2+c2) = 12" role="presentation" style="font-size: 12.8px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">12√12.2bc(b2+c2)" role="presentation" style="font-size: 12.8px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">2bc(b2+c2)−−−−−−−−−−√2bc(b2+c2)≤" role="presentation" style="font-size: 12.8px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">≤≤ 2bc22" role="presentation" style="font-size: 12.8px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">2bc22√2bc22+b222" role="presentation" style="font-size: 12.8px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">b222√b222+c222" role="presentation" style="font-size: 12.8px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">c222√c222 =(b+c)222" role="presentation" style="font-size: 12.8px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">(b+c)222√(b+c)222⇒a(b+c)bc(b2+c2)≥22a(b+c)(b+c)2=22ab+c" role="presentation" style="font-size: 16px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">⇒a(b+c)bc(b2+c2)−−−−−−−−−√≥22√a(b+c)(b+c)2=22√ab+c⇒a(b+c)bc(b2+c2)≥22a(b+c)(b+c)2=22ab+ctương tự:b(c+a)ca(c2+a2)≥22bc+a" role="presentation" style="font-size: 16px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">b(c+a)ca(c2+a2)−−−−−−−−−√≥22√bc+ab(c+a)ca(c2+a2)≥22bc+a c(a+b)ab(a2+b2)≥22ca+b" role="presentation" style="font-size: 16px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">c(a+b)ab(a2+b2)−−−−−−−−−√≥22√ca+bc(a+b)ab(a2+b2)≥22ca+bcộng cả 3 vế lại ta có: 22√ca+b22√ca+ba(b+c)bc(b2+c2)+b(c+a)ca(c2+a2)≥22bc+a+c(a+b)ab(a2+b2)≥22ca+b≥22(ab+c+bc+a+ca+b)≥22.32=32" role="presentation" style="font-size: 16px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">a(b+c)bc(b2+c2)−−−−−−−−−√+b(c+a)ca(c2+a2)−−−−−−−−−√≥22√bc+a+c(a+b)ab(a2+b2)−−−−−−−−−√≥22√ca+b≥22√(ab+c+bc+a+ca+b)≥22√.32=32

Theo BĐT AM-GM cho 2 số thực dương ta cóbc(b2+c2)" role="presentation" style="font-size: 12.8px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">bc(b2+c2)−−−−−−−−−√bc(b2+c2) = 12" role="presentation" style="font-size: 12.8px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">12√12.2bc(b2+c2)" role="presentation" style="font-size: 12.8px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">2bc(b2+c2)−−−−−−−−−−√2bc(b2+c2)≤" role="presentation" style="font-size: 12.8px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">≤≤ 2bc22" role="presentation" style="font-size: 12.8px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">2bc22√2bc22+b222" role="presentation" style="font-size: 12.8px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">b222√b222+c222" role="presentation" style="font-size: 12.8px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">c222√c222 =(b+c)222" role="presentation" style="font-size: 12.8px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">(b+c)222√(b+c)222⇒a(b+c)bc(b2+c2)≥22a(b+c)(b+c)2=22ab+c" role="presentation" style="font-size: 16px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">⇒a(b+c)bc(b2+c2)−−−−−−−−−√≥22√a(b+c)(b+c)2=22√ab+c⇒a(b+c)bc(b2+c2)≥22a(b+c)(b+c)2=22ab+ctương tự:b(c+a)ca(c2+a2)≥22bc+a" role="presentation" style="font-size: 16px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">b(c+a)ca(c2+a2)−−−−−−−−−√≥22√bc+ab(c+a)ca(c2+a2)≥22bc+a c(a+b)ab(a2+b2)≥22ca+b" role="presentation" style="font-size: 16px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">c(a+b)ab(a2+b2)−−−−−−−−−√≥22√ca+bc(a+b)ab(a2+b2)≥22ca+bcộng cả 3 vế lại ta có:$\frac{a(b+c)}{\sqrt{bc(b^{2}+c^{2})}}+\frac{b(c+a)}{\sqrt{ca(c^{2}+a^{2})}}\geq \frac{2\sqrt{2}b}{c+a} +\frac{c(a+b)}{\sqrt{ab(a^{2}+b^{2})}}\geq\frac{2\sqrt{2}c}{a+b}\geq 2\sqrt{2}(\frac{a}{b+c}+\frac{b}{c+a} + \frac{c}{a+b})\geq2\sqrt{2}.\frac{3}{2}=3\sqrt{2}$bc(b2+c2)−−−−−−−−−√+b(c+a)ca(c2+a2)−−−−−−−−−√≥22√bc+a+c(a+b)ab(a2+b2)−−−−−−−−−√≥22√ca+b≥22√(ab+c+bc+a+ca+b)≥22√.32=32bc(b2+c2)−−−−−−−−−√+b(c+a)ca(c2+a2)−−−−−−−−−√≥22√bc+a+c(a+b)ab(a2+b2)−−−−−−−−−√≥22√ca+b≥22√(ab+c+bc+a+ca+b)≥22√.32=32a(b+c)bc(b2+c2)+b(c+a)ca(c2+a2)≥22bc+a+c(a+b)ab(a2+b2)≥22ca+b≥22(ab+c+bc+a+ca+b)≥22.32=32" role="presentation" style="font-size: 16px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">a(b+c)bc(b2+c2)−−−−−−−−−√+b(c+a)ca(c2+a2)−−−−−−−−−√≥22√bc+a+c(a+b)ab(a2+b2)−−−−−−−−−√≥22√ca+b≥22√(ab+c+bc+a+ca+b)≥22√.32=32√bc(b2+c2)−−−−−−−−−√+b(c+a)ca(c2+a2)−−−−−−−−−√≥22√bc+a+c(a+b)ab(a2+b2)−−−−−−−−−√≥22√ca+b≥22√(ab+c+bc+a+ca+b)≥22√.32=32√

Theo BĐT AM-GM cho 2 số thực dương ta cóbc(b2+c2)" role="presentation" style="font-size: 12.8px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">bc(b2+c2)−−−−−−−−−√bc(b2+c2) = 12" role="presentation" style="font-size: 12.8px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">12√12.2bc(b2+c2)" role="presentation" style="font-size: 12.8px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">2bc(b2+c2)−−−−−−−−−−√2bc(b2+c2)≤" role="presentation" style="font-size: 12.8px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">≤≤ 2bc22" role="presentation" style="font-size: 12.8px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">2bc22√2bc22+b222" role="presentation" style="font-size: 12.8px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">b222√b222+c222" role="presentation" style="font-size: 12.8px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">c222√c222 =(b+c)222" role="presentation" style="font-size: 12.8px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">(b+c)222√(b+c)222⇒a(b+c)bc(b2+c2)≥22a(b+c)(b+c)2=22ab+c" role="presentation" style="font-size: 16px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">⇒a(b+c)bc(b2+c2)−−−−−−−−−√≥22√a(b+c)(b+c)2=22√ab+c⇒a(b+c)bc(b2+c2)≥22a(b+c)(b+c)2=22ab+ctương tự:b(c+a)ca(c2+a2)≥22bc+a" role="presentation" style="font-size: 16px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">b(c+a)ca(c2+a2)−−−−−−−−−√≥22√bc+ab(c+a)ca(c2+a2)≥22bc+a c(a+b)ab(a2+b2)≥22ca+b" role="presentation" style="font-size: 16px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">c(a+b)ab(a2+b2)−−−−−−−−−√≥22√ca+bc(a+b)ab(a2+b2)≥22ca+bcộng cả 3 vế lại ta có:cộng cả 3 vế lại ta có:a(b+c)bc(b2+c2)+b(c+a)ca(c2+a2)≥22bc+a+c(a+b)ab(a2+b2)≥22ca+b≥22(ab+c+bc+a+ca+b)≥22.32=32" role="presentation" style="font-size: 12.8px; position: relative;">a(b+c)bc(b2+c2)−−−−−−−−−√+b(c+a)ca(c2+a2)−−−−−−−−−√≥22√bc+a+c(a+b)ab(a2+b2)−−−−−−−−−√≥22√ca+b≥22√(ab+c+bc+a+ca+b)≥22√.32=32bc(b2+c2)−−−−−−−−−√+b(c+a)ca(c2+a2)−−−−−−−−−√≥22√bc+a+c(a+b)ab(a2+b2)−−−−−−−−−√≥22√ca+b≥22√(ab+c+bc+a+ca+b)≥22√.32=32√

Theo BĐT AM-GM cho 2 số thực dương ta cóbc(b2+c2)" role="presentation" style="font-size: 12.8px; display: inline; position: relative;">bc(b2+c2)−−−−−−−−−√bc(b2+c2) = 12" role="presentation" style="font-size: 12.8px; display: inline; position: relative;">12√12.2bc(b2+c2)" role="presentation" style="font-size: 12.8px; display: inline; position: relative;">2bc(b2+c2)−−−−−−−−−−√2bc(b2+c2)≤" role="presentation" style="font-size: 12.8px; display: inline; position: relative;">≤≤ 2bc22" role="presentation" style="font-size: 12.8px; display: inline; position: relative;">2bc22√2bc22+b222" role="presentation" style="font-size: 12.8px; display: inline; position: relative;">b222√b222+c222" role="presentation" style="font-size: 12.8px; display: inline; position: relative;">c222√c222 =(b+c)222" role="presentation" style="font-size: 12.8px; display: inline; position: relative;">(b+c)222√(b+c)222⇒a(b+c)bc(b2+c2)≥22a(b+c)(b+c)2=22ab+c" role="presentation" style="font-size: 16px; display: inline; position: relative;">⇒a(b+c)bc(b2+c2)−−−−−−−−−√≥22√a(b+c)(b+c)2=22√ab+c⇒a(b+c)bc(b2+c2)≥22a(b+c)(b+c)2=22ab+ctương tự:b(c+a)ca(c2+a2)≥22bc+a" role="presentation" style="font-size: 16px; display: inline; position: relative;">b(c+a)ca(c2+a2)−−−−−−−−−√≥22√bc+ab(c+a)ca(c2+a2)≥22bc+a c(a+b)ab(a2+b2)≥22ca+b" role="presentation" style="font-size: 16px; display: inline; position: relative;">c(a+b)ab(a2+b2)−−−−−−−−−√≥22√ca+bc(a+b)ab(a2+b2)≥22ca+bcộng cả 3 vế lại ta có:a(b+c)bc(b2+c2)+b(c+a)ca(c2+a2)≥22bc+a+c(a+b)ab(a2+b2)≥22ca+b≥22(ab+c+bc+a+ca+b)≥22.32=32" role="presentation" style="font-size: 16px; display: inline; position: relative;">a(b+c)bc(b2+c2)−−−−−−−−−√+b(c+a)ca(c2+a2)−−−−−−−−−√≥22√bc+a+c(a+b)ab(a2+b2)−−−−−−−−−√≥22√ca+b≥22√(ab+c+bc+a+ca+b)≥22√.32=32√

Theo BĐT AM-GM cho 2 số thực dương ta cóbc(b2+c2)" role="presentation" style="font-size: 12.8px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">bc(b2+c2)−−−−−−−−−√bc(b2+c2) = 12" role="presentation" style="font-size: 12.8px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">12√12.2bc(b2+c2)" role="presentation" style="font-size: 12.8px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">2bc(b2+c2)−−−−−−−−−−√2bc(b2+c2)≤" role="presentation" style="font-size: 12.8px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">≤≤ 2bc22" role="presentation" style="font-size: 12.8px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">2bc22√2bc22+b222" role="presentation" style="font-size: 12.8px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">b222√b222+c222" role="presentation" style="font-size: 12.8px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">c222√c222 =(b+c)222" role="presentation" style="font-size: 12.8px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">(b+c)222√(b+c)222⇒a(b+c)bc(b2+c2)≥22a(b+c)(b+c)2=22ab+c" role="presentation" style="font-size: 16px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">⇒a(b+c)bc(b2+c2)−−−−−−−−−√≥22√a(b+c)(b+c)2=22√ab+c⇒a(b+c)bc(b2+c2)≥22a(b+c)(b+c)2=22ab+ctương tự:b(c+a)ca(c2+a2)≥22bc+a" role="presentation" style="font-size: 16px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">b(c+a)ca(c2+a2)−−−−−−−−−√≥22√bc+ab(c+a)ca(c2+a2)≥22bc+a c(a+b)ab(a2+b2)≥22ca+b" role="presentation" style="font-size: 16px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">c(a+b)ab(a2+b2)−−−−−−−−−√≥22√ca+bc(a+b)ab(a2+b2)≥22ca+bcộng cả 3 vế lại ta có:$\frac{a(b+c)}{\sqrt{bc(b^{2}+c^{2})}}+\frac{b(c+a)}{\sqrt{ca(c^{2}+a^{2})}}\geq \frac{2\sqrt{2}b}{c+a} +\frac{c(a+b)}{\sqrt{ab(a^{2}+b^{2})}}\geq\frac{2\sqrt{2}c}{a+b}\geq 2\sqrt{2}(\frac{a}{b+c}+\frac{b}{c+a} + \frac{c}{a+b})\geq2\sqrt{2}.\frac{3}{2}=3\sqrt{2}$bc(b2+c2)−−−−−−−−−√+b(c+a)ca(c2+a2)−−−−−−−−−√≥22√bc+a+c(a+b)ab(a2+b2)−−−−−−−−−√≥22√ca+b≥22√(ab+c+bc+a+ca+b)≥22√.32=32bc(b2+c2)−−−−−−−−−√+b(c+a)ca(c2+a2)−−−−−−−−−√≥22√bc+a+c(a+b)ab(a2+b2)−−−−−−−−−√≥22√ca+b≥22√(ab+c+bc+a+ca+b)≥22√.32=32a(b+c)bc(b2+c2)+b(c+a)ca(c2+a2)≥22bc+a+c(a+b)ab(a2+b2)≥22ca+b≥22(ab+c+bc+a+ca+b)≥22.32=32" role="presentation" style="font-size: 16px; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">a(b+c)bc(b2+c2)−−−−−−−−−√+b(c+a)ca(c2+a2)−−−−−−−−−√≥22√bc+a+c(a+b)ab(a2+b2)−−−−−−−−−√≥22√ca+b≥22√(ab+c+bc+a+ca+b)≥22√.32=32√bc(b2+c2)−−−−−−−−−√+b(c+a)ca(c2+a2)−−−−−−−−−√≥22√bc+a+c(a+b)ab(a2+b2)−−−−−−−−−√≥22√ca+b≥22√(ab+c+bc+a+ca+b)≥22√.32=32√

Theo BĐT AM-GM cho 2 số thực dương ta có$\sqrt{bc(b^{2}+c^{2})}$ = $\frac{1}{\sqrt{2}}$.$\sqrt{bc(b^{2}+c^{2})}$ $\leq$ $\frac{2bc}{2\sqrt{2}}$+$\frac{b^{2}}{2\sqrt{2}}$+$\frac{c^{2}}{2\sqrt{2}}$==$\frac{(b+c)^{2}}{2\sqrt{2}}$$\Rightarrow \frac{a(b+c)}{\sqrt{bc(b^{2}+c^{2})}} \geq \frac{2\sqrt{2}a(b+c)}{(b+c)^{2}} = \frac{2\sqrt{2}a}{b+c} $tương tự:$\frac{b(c+a)}{\sqrt{ca(c^{2}+a^{2})}}\geq\frac{2\sqrt{2}b}{c+a}$ $\frac{c(a+b)}{\sqrt{ab(a^{2}+b^{2})}} \geq \frac{2\sqrt{2}c}{a+b}$cộng cả 3 vế lại ta có:$\frac{a(b+c)}{\sqrt{bc(b^{2}+c^{2})}}+\frac{b(c+a)}{\sqrt{ca(c^{2}+a^{2})}}\geq \frac{2\sqrt{2}b}{c+a} +\frac{c(a+b)}{\sqrt{ab(a^{2}+b^{2})}}\geq\frac{2\sqrt{2}c}{a+b}\geq 2\sqrt{2}(\frac{a}{b+c}+\frac{b}{c+a} + \frac{c}{a+b})\geq2\sqrt{2}.\frac{3}{2}=3\sqrt{2}$

Theo BĐT AM-GM cho 2 số thực dương ta cóbc(b2+c2)" role="presentation" style="font-size: 12.8px; display: inline; position: relative;">bc(b2+c2)−−−−−−−−−√bc(b2+c2) = 12" role="presentation" style="font-size: 12.8px; display: inline; position: relative;">12√12.2bc(b2+c2)" role="presentation" style="font-size: 12.8px; display: inline; position: relative;">2bc(b2+c2)−−−−−−−−−−√2bc(b2+c2)≤" role="presentation" style="font-size: 12.8px; display: inline; position: relative;">≤≤ 2bc22" role="presentation" style="font-size: 12.8px; display: inline; position: relative;">2bc22√2bc22+b222" role="presentation" style="font-size: 12.8px; display: inline; position: relative;">b222√b222+c222" role="presentation" style="font-size: 12.8px; display: inline; position: relative;">c222√c222 =(b+c)222" role="presentation" style="font-size: 12.8px; display: inline; position: relative;">(b+c)222√(b+c)222⇒a(b+c)bc(b2+c2)≥22a(b+c)(b+c)2=22ab+c" role="presentation" style="font-size: 16px; display: inline; position: relative;">⇒a(b+c)bc(b2+c2)−−−−−−−−−√≥22√a(b+c)(b+c)2=22√ab+c⇒a(b+c)bc(b2+c2)≥22a(b+c)(b+c)2=22ab+ctương tự:b(c+a)ca(c2+a2)≥22bc+a" role="presentation" style="font-size: 16px; display: inline; position: relative;">b(c+a)ca(c2+a2)−−−−−−−−−√≥22√bc+ab(c+a)ca(c2+a2)≥22bc+a c(a+b)ab(a2+b2)≥22ca+b" role="presentation" style="font-size: 16px; display: inline; position: relative;">c(a+b)ab(a2+b2)−−−−−−−−−√≥22√ca+bc(a+b)ab(a2+b2)≥22ca+bcộng cả 3 vế lại ta có:a(b+c)bc(b2+c2)+b(c+a)ca(c2+a2)≥22bc+a+c(a+b)ab(a2+b2)≥22ca+b≥22(ab+c+bc+a+ca+b)≥22.32=32" role="presentation" style="font-size: 16px; display: inline; position: relative;">a(b+c)bc(b2+c2)−−−−−−−−−√+b(c+a)ca(c2+a2)−−−−−−−−−√≥22√bc+a+c(a+b)ab(a2+b2)−−−−−−−−−√≥22√ca+b≥22√(ab+c+bc+a+ca+b)≥22√.32=32√

Theo BĐT AM-GM cho 2 số thực dương ta có$\sqrt{bc(b^{2}+c^{2})}$=$\frac{1}{\sqrt{2}}$.$\sqrt{2bc(b^{2}+c^{2})}$ $\leq$ $\frac{2bc}{2\sqrt{2}}$+$\frac{b^{2}}{2\sqrt{2}}$+$\frac{c^{2}}{2\sqrt{2}}$=$\frac{(b+c)^{2}}{2\sqrt{2}}$$\Rightarrow \frac{a(b+c)}{\sqrt{bc(b^{2}+c^{2})}} \geq \frac{2\sqrt{2}a(b+c)}{(b+c)^{2}} = \frac{2\sqrt{2}a}{b+c} $tương tự:$\frac{b(c+a)}{\sqrt{ca(c^{2}+a^{2})}}\geq\frac{2\sqrt{2}b}{c+a}$ $\frac{c(a+b)}{\sqrt{ab(a^{2}+b^{2})}} \geq \frac{2\sqrt{2}c}{a+b}$cộng cả 3 vế lại ta có:$\frac{a(b+c)}{\sqrt{bc(b^{2}+c^{2})}}+\frac{b(c+a)}{\sqrt{ca(c^{2}+a^{2})}}\geq \frac{2\sqrt{2}b}{c+a} +\frac{c(a+b)}{\sqrt{ab(a^{2}+b^{2})}}\geq\frac{2\sqrt{2}c}{a+b}\geq 2\sqrt{2}(\frac{a}{b+c}+\frac{b}{c+a} + \frac{c}{a+b})\geq2\sqrt{2}.\frac{3}{2}=3\sqrt{2}$

Theo BĐT AM-GM cho 2 số thực dương ta có$\sqrt{bc(b^{2}+c^{2})}$ = $\frac{1}{\sqrt{2}}$.$\sqrt{bc(b^{2}+c^{2})}$ $\leq$ $\frac{2bc}{2\sqrt{2}}$+$\frac{b^{2}}{2\sqrt{2}}$+$\frac{c^{2}}{2\sqrt{2}}$==$\frac{(b+c)^{2}}{2\sqrt{2}}$$\Rightarrow \frac{a(b+c)}{\sqrt{bc(b^{2}+c^{2})}} \geq \frac{2\sqrt{2}a(b+c)}{(b+c)^{2}} = \frac{2\sqrt{2}a}{b+c} $tương tự:$\frac{b(c+a)}{\sqrt{ca(c^{2}+a^{2})}}\geq\frac{2\sqrt{2}b}{c+a}$ $\frac{c(a+b)}{\sqrt{ab(a^{2}+b^{2})}} \geq \frac{2\sqrt{2}c}{a+b}$cộng cả 3 vế lại ta có:$\frac{a(b+c)}{\sqrt{bc(b^{2}+c^{2})}}+\frac{b(c+a)}{\sqrt{ca(c^{2}+a^{2})}}\geq \frac{2\sqrt{2}b}{c+a} +\frac{c(a+b)}{\sqrt{ab(a^{2}+b^{2})}}\geq\frac{2\sqrt{2}c}{a+b}\geq 2\sqrt{2}(\frac{a}{b+c}+\frac{b}{c+a} + \frac{c}{a+b})\geq2\sqrt{2}.\frac{3}{2}=3\sqrt{2}$

Theo BĐT AM-GM cho 2 số thực dương ta có$\sqrt{bc(b^{2}+c^{2})}=\frac{1}{\sqrt{2}}.\sqrt{2bc(b^{2}+c^{2})}$ $\leq \frac{2bc}{2\sqrt{2}}+\frac{b^{2}}{2\sqrt{2}}+\frac{c^{2}}{2\sqrt{2}}$=$\frac{(b+c)^{2}}{2\sqrt{2}}$$\Rightarrow \frac{a(b+c)}{\sqrt{bc(b^{2}+c^{2})}} \geq \frac{2\sqrt{2}a(b+c)}{(b+c)^{2}} = \frac{2\sqrt{2}a}{b+c} $tương tự:$\frac{b(c+a)}{\sqrt{ca(c^{2}+a^{2})}}\geq\frac{2\sqrt{2}b}{c+a}$ $\frac{c(a+b)}{\sqrt{ab(a^{2}+b^{2})}} \geq \frac{2\sqrt{2}c}{a+b}$cộng cả 3 vế lại ta có:$\frac{a(b+c)}{\sqrt{bc(b^{2}+c^{2})}}+\frac{b(c+a)}{\sqrt{ca(c^{2}+a^{2})}}\geq \frac{2\sqrt{2}b}{c+a} +\frac{c(a+b)}{\sqrt{ab(a^{2}+b^{2})}}\geq\frac{2\sqrt{2}c}{a+b}\geq 2\sqrt{2}(\frac{a}{b+c}+\frac{b}{c+a} + \frac{c}{a+b})\geq2\sqrt{2}.\frac{3}{2}=3\sqrt{2}$

Theo BĐT AM-GM cho 2 số thực dương ta có$\sqrt{bc(b^{2}+c^{2})}$=$\frac{1}{\sqrt{2}}$.$\sqrt{2bc(b^{2}+c^{2})}$ $\leq$ $\frac{2bc}{2\sqrt{2}}$+$\frac{b^{2}}{2\sqrt{2}}$+$\frac{c^{2}}{2\sqrt{2}}$=$\frac{(b+c)^{2}}{2\sqrt{2}}$$\Rightarrow \frac{a(b+c)}{\sqrt{bc(b^{2}+c^{2})}} \geq \frac{2\sqrt{2}a(b+c)}{(b+c)^{2}} = \frac{2\sqrt{2}a}{b+c} $tương tự:$\frac{b(c+a)}{\sqrt{ca(c^{2}+a^{2})}}\geq\frac{2\sqrt{2}b}{c+a}$ $\frac{c(a+b)}{\sqrt{ab(a^{2}+b^{2})}} \geq \frac{2\sqrt{2}c}{a+b}$cộng cả 3 vế lại ta có:$\frac{a(b+c)}{\sqrt{bc(b^{2}+c^{2})}}+\frac{b(c+a)}{\sqrt{ca(c^{2}+a^{2})}}\geq \frac{2\sqrt{2}b}{c+a} +\frac{c(a+b)}{\sqrt{ab(a^{2}+b^{2})}}\geq\frac{2\sqrt{2}c}{a+b}\geq 2\sqrt{2}(\frac{a}{b+c}+\frac{b}{c+a} + \frac{c}{a+b})\geq2\sqrt{2}.\frac{3}{2}=3\sqrt{2}$

Theo BĐT AM-GM cho 2 số thực dương ta có$\sqrt{bc(b^{2}+c^{2})}$= $\frac{1}{\sqrt{2}}$. $\sqrt{2bc(b^{2}+c^{2})}$ $\leq$ $\frac{2bc}{2\sqrt{2}}$+$\frac{b^{2}}{2\sqrt{2}}$+$\frac{c^{2}}{2\sqrt{2}}$=$\frac{(b+c)^{2}}{2\sqrt{2}}$$\Rightarrow \frac{a(b+c)}{\sqrt{bc(b^{2}+c^{2})}} \geq \frac{2\sqrt{2}a(b+c)}{(b+c)^{2}} = \frac{2\sqrt{2}a}{b+c} $tương tự:$\frac{b(c+a)}{\sqrt{ca(c^{2}+a^{2})}}\geq\frac{2\sqrt{2}b}{c+a}$ $\frac{c(a+b)}{\sqrt{ab(a^{2}+b^{2})}} \geq \frac{2\sqrt{2}c}{a+b}$cộng cả 3 vế lại ta có:$\frac{a(b+c)}{\sqrt{bc(b^{2}+c^{2})}}+\frac{b(c+a)}{\sqrt{ca(c^{2}+a^{2})}}\geq \frac{2\sqrt{2}b}{c+a} +\frac{c(a+b)}{\sqrt{ab(a^{2}+b^{2})}}\geq\frac{2\sqrt{2}c}{a+b}\geq 2\sqrt{2}(\frac{a}{b+c}+\frac{b}{c+a} + \frac{c}{a+b})\geq2\sqrt{2}.\frac{3}{2}=3\sqrt{2}$

Theo BĐT AM-GM cho 2 số thực dương ta có$\sqrt{bc(b^{2}+c^{2})}=\frac{1}{\sqrt{2}}.\sqrt{2bc(b^{2}+c^{2})}$ $\leq \frac{2bc}{2\sqrt{2}}+\frac{b^{2}}{2\sqrt{2}}+\frac{c^{2}}{2\sqrt{2}}$=$\frac{(b+c)^{2}}{2\sqrt{2}}$$\Rightarrow \frac{a(b+c)}{\sqrt{bc(b^{2}+c^{2})}} \geq \frac{2\sqrt{2}a(b+c)}{(b+c)^{2}} = \frac{2\sqrt{2}a}{b+c} $tương tự:$\frac{b(c+a)}{\sqrt{ca(c^{2}+a^{2})}}\geq\frac{2\sqrt{2}b}{c+a}$ $\frac{c(a+b)}{\sqrt{ab(a^{2}+b^{2})}} \geq \frac{2\sqrt{2}c}{a+b}$cộng cả 3 vế lại ta có:$\frac{a(b+c)}{\sqrt{bc(b^{2}+c^{2})}}+\frac{b(c+a)}{\sqrt{ca(c^{2}+a^{2})}}\geq \frac{2\sqrt{2}b}{c+a} +\frac{c(a+b)}{\sqrt{ab(a^{2}+b^{2})}}\geq\frac{2\sqrt{2}c}{a+b}\geq 2\sqrt{2}(\frac{a}{b+c}+\frac{b}{c+a} + \frac{c}{a+b})\geq2\sqrt{2}.\frac{3}{2}=3\sqrt{2}$

Theo BĐT AM-GM cho 2 số thực dương ta có$\sqrt{bc(b^{2}+c^{2})}$=$\frac{1}{\sqrt{2}}$. $\sqrt{2bc(b^{2}+c^{2})}$ $\leq$ $\frac{2bc}{2\sqrt{2}}$+$\frac{b^{2}}{2\sqrt{2}}$+$\frac{c^{2}}{2\sqrt{2}}$=$\frac{(b+c)^{2}}{2\sqrt{2}}$$\Rightarrow \frac{a(b+c)}{\sqrt{bc(b^{2}+c^{2})}} \geq \frac{2\sqrt{2}a(b+c)}{(b+c)^{2}} = \frac{2\sqrt{2}a}{b+c} $tương tự:$\frac{b(c+a)}{\sqrt{ca(c^{2}+a^{2})}}\geq\frac{2\sqrt{2}b}{c+a}$ $\frac{c(a+b)}{\sqrt{ab(a^{2}+b^{2})}} \geq \frac{2\sqrt{2}c}{a+b}$cộng cả 3 vế lại ta có:$\frac{a(b+c)}{\sqrt{bc(b^{2}+c^{2})}}+\frac{b(c+a)}{\sqrt{ca(c^{2}+a^{2})}}\geq \frac{2\sqrt{2}b}{c+a} +\frac{c(a+b)}{\sqrt{ab(a^{2}+b^{2})}}\geq\frac{2\sqrt{2}c}{a+b}\geq 2\sqrt{2}(\frac{a}{b+c}+\frac{b}{c+a} + \frac{c}{a+b})\geq2\sqrt{2}.\frac{3}{2}=3\sqrt{2}$

Theo BĐT AM-GM cho 2 số thực dương ta có$\sqrt{bc(b^{2}+c^{2})}$= $\frac{1}{\sqrt{2}}$. $\sqrt{2bc(b^{2}+c^{2})}$ $\leq$ $\frac{2bc}{2\sqrt{2}}$+$\frac{b^{2}}{2\sqrt{2}}$+$\frac{c^{2}}{2\sqrt{2}}$=$\frac{(b+c)^{2}}{2\sqrt{2}}$$\Rightarrow \frac{a(b+c)}{\sqrt{bc(b^{2}+c^{2})}} \geq \frac{2\sqrt{2}a(b+c)}{(b+c)^{2}} = \frac{2\sqrt{2}a}{b+c} $tương tự:$\frac{b(c+a)}{\sqrt{ca(c^{2}+a^{2})}}\geq\frac{2\sqrt{2}b}{c+a}$ $\frac{c(a+b)}{\sqrt{ab(a^{2}+b^{2})}} \geq \frac{2\sqrt{2}c}{a+b}$cộng cả 3 vế lại ta có:$\frac{a(b+c)}{\sqrt{bc(b^{2}+c^{2})}}+\frac{b(c+a)}{\sqrt{ca(c^{2}+a^{2})}}\geq \frac{2\sqrt{2}b}{c+a} +\frac{c(a+b)}{\sqrt{ab(a^{2}+b^{2})}}\geq\frac{2\sqrt{2}c}{a+b}\geq 2\sqrt{2}(\frac{a}{b+c}+\frac{b}{c+a} + \frac{c}{a+b})\geq2\sqrt{2}.\frac{3}{2}=3\sqrt{2}$

Theo BĐT AM-GM cho 2 số thực dương ta có$\sqrt{bc(b^{2}+c^{2})} = \frac{1}{\sqrt{2}}.\sqrt{2bc(b^{2}+c^{2})} \leq \frac{2bc+b^{2}+c^{2}}{2\sqrt{2}} =\frac{(b+c)^{2}}{2\sqrt{2}}$$\Rightarrow \frac{a(b+c)}{\sqrt{bc(b^{2}+c^{2})}} \geq \frac{2\sqrt{2}a(b+c)}{(b+c)^{2}} = \frac{2\sqrt{2}a}{b+c} $tương tự:$\frac{b(c+a)}{\sqrt{ca(c^{2}+a^{2})}}\geq\frac{2\sqrt{2}b}{c+a}$ $\frac{c(a+b)}{\sqrt{ab(a^{2}+b^{2})}} \geq \frac{2\sqrt{2}c}{a+b}$cộng cả 3 vế lại ta có:$\frac{a(b+c)}{\sqrt{bc(b^{2}+c^{2})}}+\frac{b(c+a)}{\sqrt{ca(c^{2}+a^{2})}}\geq \frac{2\sqrt{2}b}{c+a} +\frac{c(a+b)}{\sqrt{ab(a^{2}+b^{2})}}\geq\frac{2\sqrt{2}c}{a+b}\geq 2\sqrt{2}(\frac{a}{b+c}+\frac{b}{c+a} + \frac{c}{a+b})\geq2\sqrt{2}.\frac{3}{2}=3\sqrt{2}$

Theo BĐT AM-GM cho 2 số thực dương ta có$\sqrt{bc(b^{2}+c^{2})}$=$\frac{1}{\sqrt{2}}$. $\sqrt{2bc(b^{2}+c^{2})}$ $\leq$ $\frac{2bc}{2\sqrt{2}}$+$\frac{b^{2}}{2\sqrt{2}}$+$\frac{c^{2}}{2\sqrt{2}}$=$\frac{(b+c)^{2}}{2\sqrt{2}}$$\Rightarrow \frac{a(b+c)}{\sqrt{bc(b^{2}+c^{2})}} \geq \frac{2\sqrt{2}a(b+c)}{(b+c)^{2}} = \frac{2\sqrt{2}a}{b+c} $tương tự:$\frac{b(c+a)}{\sqrt{ca(c^{2}+a^{2})}}\geq\frac{2\sqrt{2}b}{c+a}$ $\frac{c(a+b)}{\sqrt{ab(a^{2}+b^{2})}} \geq \frac{2\sqrt{2}c}{a+b}$cộng cả 3 vế lại ta có:$\frac{a(b+c)}{\sqrt{bc(b^{2}+c^{2})}}+\frac{b(c+a)}{\sqrt{ca(c^{2}+a^{2})}}\geq \frac{2\sqrt{2}b}{c+a} +\frac{c(a+b)}{\sqrt{ab(a^{2}+b^{2})}}\geq\frac{2\sqrt{2}c}{a+b}\geq 2\sqrt{2}(\frac{a}{b+c}+\frac{b}{c+a} + \frac{c}{a+b})\geq2\sqrt{2}.\frac{3}{2}=3\sqrt{2}$

Theo BĐT AM-GM cho 2 số thực dương ta có$ \sqrt{bc(b^{2}+c^{2})} = \frac{1}{\sqrt{2}}. \sqrt{2bc(b^{2}+c^{2})} \leq \frac{2bc+b^{2}+c^{2}}{2\sqrt{2}} =\frac{(b+c)^{2}}{2\sqrt{2}}\Rightarrow \frac{a(b+c)}{\sqrt{bc(b^{2}+c^{2})}} \geq \frac{2\sqrt{2}a(b+c)}{(b+c)^{2}} = \frac{2\sqrt{2}a}{b+c} $tương tự:$\frac{b(c+a)}{\sqrt{ca(c^{2}+a^{2})}} \geq \frac{2\sqrt{2}b}{c+a} \frac{c(a+b)}{\sqrt{ab(a^{2}+b^{2})}} \geq \frac{2\sqrt{2}c}{a+b}$cộng cả 3 vế lại ta có:$\frac{a(b+c)}{\sqrt{bc(b^{2}+c^{2})}} + \frac{b(c+a)}{\sqrt{ca(c^{2}+a^{2})}} \geq \frac{2\sqrt{2}b}{c+a} + \frac{c(a+b)}{\sqrt{ab(a^{2}+b^{2})}} \geq \frac{2\sqrt{2}c}{a+b} \geq 2\sqrt{2}(\frac{a}{b+c} + \frac{b}{c+a} + \frac{c}{a+b}) \geq 2\sqrt{2}.\frac{3}{2} = 3\sqrt{2} $

Theo BĐT AM-GM cho 2 số thực dương ta có$\sqrt{bc(b^{2}+c^{2})} = \frac{1}{\sqrt{2}}.\sqrt{2bc(b^{2}+c^{2})} \leq \frac{2bc+b^{2}+c^{2}}{2\sqrt{2}} =\frac{(b+c)^{2}}{2\sqrt{2}}$$\Rightarrow \frac{a(b+c)}{\sqrt{bc(b^{2}+c^{2})}} \geq \frac{2\sqrt{2}a(b+c)}{(b+c)^{2}} = \frac{2\sqrt{2}a}{b+c} $tương tự:$\frac{b(c+a)}{\sqrt{ca(c^{2}+a^{2})}}\geq\frac{2\sqrt{2}b}{c+a}$ $\frac{c(a+b)}{\sqrt{ab(a^{2}+b^{2})}} \geq \frac{2\sqrt{2}c}{a+b}$cộng cả 3 vế lại ta có:$\frac{a(b+c)}{\sqrt{bc(b^{2}+c^{2})}}+\frac{b(c+a)}{\sqrt{ca(c^{2}+a^{2})}}\geq \frac{2\sqrt{2}b}{c+a} +\frac{c(a+b)}{\sqrt{ab(a^{2}+b^{2})}}\geq\frac{2\sqrt{2}c}{a+b}\geq 2\sqrt{2}(\frac{a}{b+c}+\frac{b}{c+a} + \frac{c}{a+b})\geq2\sqrt{2}.\frac{3}{2}=3\sqrt{2}$

Theo BĐT AM-GM cho 2 số thực dương ta có$ \sqrt{bc(b^{2}+c^{2})} = \frac{1}{\sqrt{2}}. \sqrt{2bc(b^{2}+c^{2})} \leq \frac{2bc+b^{2}+c^{2}}{2\sqrt{2}} =\frac{(b+c)^{2}}{2\sqrt{2}}\Rightarrow \frac{a(b+c)}{\sqrt{bc(b^{2}+c^{2})}} \geq \frac{2\sqrt{2}a(b+c)}{(b+c)^{2}} = \frac{2\sqrt{2}a}{b+c} $tương tự:$\frac{b(c+a)}{\sqrt{ca(c^{2}+a^{2})}} \geq \frac{2\sqrt{2}b}{c+a} \frac{c(a+b)}{\sqrt{ab(a^{2}+b^{2})}} \geq \frac{2\sqrt{2}c}{a+b}$cộng cả 3 vế lại ta có:$\frac{a(b+c)}{\sqrt{bc(b^{2}+c^{2})}} + \frac{b(c+a)}{\sqrt{ca(c^{2}+a^{2})}} \geq \frac{2\sqrt{2}b}{c+a} + \frac{c(a+b)}{\sqrt{ab(a^{2}+b^{2})}} \geq \frac{2\sqrt{2}c}{a+b} \geq 2\sqrt{2}(\frac{a}{b+c} + \frac{b}{c+a} + \frac{c}{a+b}) \geq 2\sqrt{2}.\frac{3}{2} = 3\sqrt{2} $

Theo BĐT AM-GM cho 2 số thực dương ta có$ \sqrt{bc(b^{2}+c^{2})} = \frac{1}{\sqrt{2}}. \sqrt{2bc(b^{2}+c^{2})} \leq \frac{2bc+b^{2}+c^{2}}{2\sqrt{2}} =\frac{(b+c)^{2}}{2\sqrt{2}}\Rightarrow \frac{a(b+c)}{\sqrt{bc(b^{2}+c^{2})}} \geq \frac{2\sqrt{2}a(b+c)}{(b+c)^{2}} = \frac{2\sqrt{2}a}{b+c} $tương tự:$\frac{b(c+a)}{\sqrt{ca(c^{2}+a^{2})}} \geq \frac{2\sqrt{2}b}{c+a} \frac{c(a+b)}{\sqrt{ab(a^{2}+b^{2})}} \geq \frac{2\sqrt{2}c}{a+b}$cộng cả 3 vế lại ta có:$\frac{a(b+c)}{\sqrt{bc(b^{2}+c^{2})}} + \frac{b(c+a)}{\sqrt{ca(c^{2}+a^{2})}} \geq \frac{2\sqrt{2}b}{c+a} + \frac{c(a+b)}{\sqrt{ab(a^{2}+b^{2})}} \geq \frac{2\sqrt{2}c}{a+b} \geq 2\sqrt{2}(\frac{a}{b+c} + \frac{b}{c+a} + \frac{c}{a+b}) \geq 2\sqrt{2}.\frac{3}{2} = 3\sqrt{2} $

Theo BĐT AM-GM cho 2 số thực dương ta có$ \sqrt{bc(b^{2}+c^{2})} = \frac{1}{\sqrt{2}}. \sqrt{2bc(b^{2}+c^{2})} \leq \frac{2bc+b^{2}+c^{2}}{2\sqrt{2}} =\frac{(b+c)^{2}}{2\sqrt{2}}\Rightarrow \frac{a(b+c)}{\sqrt{bc(b^{2}+c^{2})}} \geq \frac{2\sqrt{2}a(b+c)}{(b+c)^{2}} = \frac{2\sqrt{2}a}{b+c} tương tự:\frac{b(c+a)}{\sqrt{ca(c^{2}+a^{2})}} \geq \frac{2\sqrt{2}b}{c+a} \frac{c(a+b)}{\sqrt{ab(a^{2}+b^{2})}} \geq \frac{2\sqrt{2}c}{a+b}cộng cả 3 vế lại ta có:\frac{a(b+c)}{\sqrt{bc(b^{2}+c^{2})}} + \frac{b(c+a)}{\sqrt{ca(c^{2}+a^{2})}} \geq \frac{2\sqrt{2}b}{c+a} + \frac{c(a+b)}{\sqrt{ab(a^{2}+b^{2})}} \geq \frac{2\sqrt{2}c}{a+b} \geq 2\sqrt{2}(\frac{a}{b+c} + \frac{b}{c+a} + \frac{c}{a+b}) \geq 2\sqrt{2}.\frac{3}{2} = 3\sqrt{2} $

Theo BĐT AM-GM cho 2 số thực dương ta có$ \sqrt{bc(b^{2}+c^{2})} = \frac{1}{\sqrt{2}}. \sqrt{2bc(b^{2}+c^{2})} \leq \frac{2bc+b^{2}+c^{2}}{2\sqrt{2}} =\frac{(b+c)^{2}}{2\sqrt{2}}\Rightarrow \frac{a(b+c)}{\sqrt{bc(b^{2}+c^{2})}} \geq \frac{2\sqrt{2}a(b+c)}{(b+c)^{2}} = \frac{2\sqrt{2}a}{b+c} $tương tự:$\frac{b(c+a)}{\sqrt{ca(c^{2}+a^{2})}} \geq \frac{2\sqrt{2}b}{c+a} \frac{c(a+b)}{\sqrt{ab(a^{2}+b^{2})}} \geq \frac{2\sqrt{2}c}{a+b}$cộng cả 3 vế lại ta có:$\frac{a(b+c)}{\sqrt{bc(b^{2}+c^{2})}} + \frac{b(c+a)}{\sqrt{ca(c^{2}+a^{2})}} \geq \frac{2\sqrt{2}b}{c+a} + \frac{c(a+b)}{\sqrt{ab(a^{2}+b^{2})}} \geq \frac{2\sqrt{2}c}{a+b} \geq 2\sqrt{2}(\frac{a}{b+c} + \frac{b}{c+a} + \frac{c}{a+b}) \geq 2\sqrt{2}.\frac{3}{2} = 3\sqrt{2} $