Question

\(\dfrac{\sqrt{ab}}{\left(a+1\right)\left(b+1\right)}\le\dfrac{1}{4}\)với a,b>0

Answer 1

\(VT=\dfrac{\sqrt{ab}}{\left(a+1\right)\left(b+1\right)}\le\dfrac{\sqrt{ab}}{2\sqrt{a}\cdot2\sqrt{b}}=\dfrac{1}{4}=VP\)

\("="\Leftrightarrow a=b=1\)

Answer 2

Lời giải:

Áp dụng BĐT Cô -si cho hai số \(a,b>0\) ta có:

\(\left\{\begin{matrix} a+1\geq 2\sqrt{a}\\ b+1\geq 2\sqrt{b}\end{matrix}\right.\)

\(\Rightarrow (a+1)(b+1)\geq 4\sqrt{ab}\)

\(\Rightarrow \frac{\sqrt{ab}}{(a+1)(b+1)}\leq \frac{\sqrt{ab}}{4\sqrt{ab}}=\frac{1}{4}\)

Ta có đpcm.

Dấu bằng xảy ra khi \(a=b=1\)