Question

Cho a,b,c>0 TM `2a^2+b^2+c^2=4`

Tìm `min_T=(b+1)/((a+c)^2+4abc)+(c+1)/((a+b)^2+4abc)`

Answer 1

\(4b.ac+\left(a+c\right)^2\le4b.\dfrac{1}{4}\left(a+c\right)^2+\left(a+c\right)^2=\left(a+c\right)^2\left(b+1\right)\)

\(\Rightarrow T\ge\dfrac{1}{\left(a+c\right)^2}+\dfrac{1}{\left(a+b\right)^2}\ge\dfrac{1}{2\left(a^2+c^2\right)}+\dfrac{1}{2\left(a^2+b^2\right)}\ge\dfrac{4}{2\left(2a^2+b^2+c^2\right)}\)