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√(a^2 +b^2) - √(c^2+d^2) ≤ √((a-c)^2 + (b-d)^2)
=> a^2+b^2+c^2+d^2 - 2.√((a^2+b^2)(c^2+d^2)) ≤ a^2 + b^2 + c^2 + d^2 - 2(ac+bd)
ta cần c/m √((a^2+b^2)(c^2+d^2))) ≥ (ac+bd) <=> (a^2+b^2)(c^2+d^2) ≥ (ac+bd)^2
<=> a^2c^2 + a^2d^2+b^2c^2+b^2d^2 ≥ a^2c^2+b^2d^2+2abcd <=> (ad - bc)^2 ≥ 0 (luôn đúng)
Áp dụng: √(x^2-4x+5) - √(x^2+6x+13) ≤ √[(x-2)^2 + 1^2] - √[(x+3)^2 + 2^2] ≤ √[(x-2-x-3)^2 + (1-2)^2 = √26
Dấu "=" xảy ra <=> 2(x-2) = 1(x+3) <=> 2x - 4 = x + 3 <=> x=7
Vậy Max= √26 khi x=7
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