Question

Cho \(x,y,z\ge0\).Tìm giá trị lớn nhất :

\(P=\dfrac{x}{x^2+y^2+2}+\dfrac{y}{y^2+z^2+2}+\dfrac{z}{z^2+x^2+2}\)

Answer 1

Ta có: \(\dfrac{x}{x^2+1+y^2+1}\le\dfrac{x}{2\sqrt{\left(x^2+1\right)\left(y^2+1\right)}}\le\dfrac{1}{4}\left(\dfrac{x^2}{x^2+1}+\dfrac{1}{y^2+1}\right)\)

Tương tự: \(\dfrac{y}{y^2+z^2+2}\le\dfrac{1}{4}\left(\dfrac{y^2}{y^2+1}+\dfrac{1}{z^2+1}\right)\) ; \(\dfrac{z}{z^2+x^2+2}\le\dfrac{1}{4}\left(\dfrac{z^2}{z^2+1}+\dfrac{1}{x^2+1}\right)\)

Cộng vế với vế:

\(P\le\dfrac{1}{4}\left(\dfrac{x^2}{x^2+1}+\dfrac{1}{x^2+1}+\dfrac{y^2}{y^2+1}+\dfrac{1}{y^2+1}+\dfrac{z^2}{z^2+1}+\dfrac{1}{z^2+1}\right)=\dfrac{3}{4}\)

Dấu "=" xảy ra khi \(x=y=z=1\)