Question

Cho x,y \(\in\)[0,1] .CMR

a,\(\left(1+x\right)^2\)\(\ge\)4\(x^2\)

b,\(\left(1+x+y\right)^2\)\(\ge\)4\(\left(x^2+y^2\right)\)

Answer 1

a.

Do \(0\le x\le1\Rightarrow\left(1+x\right)^2\ge\left(x+x\right)^2=4x^2\) (đpcm)

Dấu "=" xảy ra khi \(x=1\)

b.

Do \(x;y\in\left[0;1\right]\Rightarrow\left\{{}\begin{matrix}x^2\le x\\y^2\le y\end{matrix}\right.\) \(\Rightarrow x+y\ge x^2+y^2\)

\(\Rightarrow\left(1+x+y\right)^2\ge4\left(x+y\right)\ge4\left(x^2+y^2\right)\) (đpcm)

Dấu "=" xảy ra khi \(\left(x;y\right)=\left(1;0\right);\left(0;1\right)\)