chứng minh $\frac{7x+5y}{\sqrt{140}}\geq\sqrt{xy}$

Question

chứng minh $\frac{7x+5y}{\sqrt{140}}\geq\sqrt{xy}$ ($x\geq0,y\geq0$)

score 4 · Answer 1

+ Nếu $y=0$ thì BĐT luôn đúng.
+ Nếu $y>0$ BĐT
$\Leftrightarrow \frac{7}{\sqrt{140}}\frac{x}{y}-\sqrt{\frac{x}{y}}+\frac{5}{\sqrt{140}} \ge 0$
$\Leftrightarrow f(t)=\frac{7}{\sqrt{140}}t^2-t+\frac{5}{\sqrt{140}} \ge 0$, với $t=\sqrt{\frac{x}{y}}.$
Ta có $\Delta_f=1-4.\frac{7}{\sqrt{140}}.\frac{5}{\sqrt{140}}=0\Rightarrow f(t)\ge0 \quad \forall t \ge 0.$