Question

Cho 0<x<1; 0<y<1. CMR:

\(x+y+x\sqrt{1-y^2}+y\sqrt{1-x^2}\le\frac{3\sqrt{3}}{2}\)

Answer 1

\(f\left(x;y\right)=x+y+x\sqrt{1-y^2}+y\sqrt{1-x^2}\)

\(\Rightarrow\frac{\sqrt{3}}{2}f\left(x;y\right)=\frac{\sqrt{3}}{2}\left(x+y\right)+\frac{1}{2}\left(x\sqrt{3-3y^2}+y\sqrt{3-3x^2}\right)\)

\(\Rightarrow\frac{\sqrt{3}}{2}f\left(x;y\right)\le\frac{\frac{3}{4}+x^2+\frac{3}{4}+y^2}{2}+\frac{1}{2}\left(\frac{-3x^2+y^2+3-3y^2+x^2+3}{2}\right)\)

\(\Rightarrow\frac{\sqrt{3}}{2}f\left(x;y\right)\le\frac{\frac{3}{2}+x^2+y^2-x^2-y^2+3}{2}=\frac{9}{4}\)

\(\Rightarrow f\left(x;y\right)\le\frac{3\sqrt{3}}{2}\)

Dấu "=" khi x = y = \(\frac{\sqrt{3}}{2}\).

#Kaito#