Question

Cho\(\left\{{}\begin{matrix}a,b,c>0\\a+b+c\le1\end{matrix}\right.\). cmr: \(\dfrac{1}{a^2}\)+\(\dfrac{1}{b^2}\)+\(\dfrac{1}{c^2}\)+\(\dfrac{2}{ab}\)+\(\dfrac{2}{bc}\)+\(\dfrac{2}{ac}\) \(\ge\)81

Answer 1

Ta có :

\(VT=\dfrac{1}{a^2}+\dfrac{1}{b^2}+\dfrac{1}{c^2}+\dfrac{2}{ab}+\dfrac{2}{bc}+\dfrac{2}{ca}=\dfrac{1}{a^2}+\dfrac{1}{b^2}+\dfrac{1}{c^2}+\dfrac{4}{2ab}+\dfrac{4}{2bc}+\dfrac{4}{2ca}\)

Theo BĐT Cauchy schwarz dưới dạng engel ta có :

\(VT\ge\dfrac{\left(1+1+1+2+2+2\right)^2}{\left(a+b+c\right)^2}=\dfrac{81}{1}=81\)

Vậy BĐT đã được chứng minh . Dấu \("="\) xảy ra khi \(a=b=c=\dfrac{1}{3}\)