Question

Cho a, b, c > 4.

Tìm MIN  M =  \(\frac{a}{\sqrt{b}-2}+\frac{b}{\sqrt{c}-2}+\frac{c}{\sqrt{a}-2}\)

Answer 1

Dự đoán dấu "=" xảy ra khi \(a=b=c=16\Rightarrow M=24\)

Ta cần cm \(M=24\) là GTNN

Áp dụng BĐT Cauchy-Schwarz dạng Engel ta có:

\(M=\frac{a}{\sqrt{b}-2}+\frac{b}{\sqrt{c}-2}+\frac{c}{\sqrt{a}-2}\)

\(\ge\frac{\left(\sqrt{a}+\sqrt{b}+\sqrt{c}\right)^2}{\sqrt{a}+\sqrt{b}+\sqrt{c}-6}\). Tức là cần cm \(\frac{\left(\sqrt{a}+\sqrt{b}+\sqrt{c}\right)^2}{\sqrt{a}+\sqrt{b}+\sqrt{c}-6}\ge24\)

\(\Leftrightarrow\frac{t^2}{t-6}\ge24\forall\left(t=\sqrt{a}+\sqrt{b}+\sqrt{c}>6\right)\)

\(\Leftrightarrow t^2\ge24\left(t-6\right)\Leftrightarrow t^2-24t+144\ge0\Leftrightarrow\left(t-12\right)^2\ge0\)*Đúng*