Question

2) Cho a,b ≥ 0 .CMR :

a) \(\dfrac{a^2+b^2}{2}\) ≥ \(\left(\dfrac{a+b}{2}\right)^2\) b) \(\dfrac{a^3+b^3}{2}\) ≥ \(\left(\dfrac{a+b}{2}\right)^3\)

Answer 1

Theo cách lớp 8 :vvv

Câu a : \(\dfrac{a^2+b^2}{2}\ge\left(\dfrac{a+b}{2}\right)^2\)

\(\Leftrightarrow4a^2+4b^2\ge2a^2+2b^2+4ab\)

\(\Leftrightarrow2\left(a-b\right)^2\ge0\) ( Đúng )

Dấu \("="\)xảy ra khi \(a=b\)

Câu b : \(\dfrac{a^3+b^3}{2}\ge\left(\dfrac{a+b}{2}\right)^3\)

\(\Leftrightarrow8a^3+8b^3\ge2a^3+2b^3+6a^2b+6ab^2\)

\(\Leftrightarrow6a^3-6a^2b+6b^3-6b^2a\ge0\)

\(\Leftrightarrow6a^2\left(a-b\right)-6b^2\left(a-b\right)\ge0\)

\(\Leftrightarrow6\left(a-b\right)^2\left(a+b\right)\ge0\) ( Đúng )

Dấu \("="\) xảy ra khi \(\left[{}\begin{matrix}a=b\\a=-b\end{matrix}\right.\)