Question

[Uchiha Itachi]

Cho 0<x,y,z<1; xy+yz+zx+2xyz=1
Tìm GTLN của P=\(\sqrt{1-x^2}+\sqrt{1-y^2}+\sqrt{1-z^2}\)
Mọi người giúp mình với ạ, mình đang cần gấp ạ.

Answer 1

[Nguyễn Việt Lâm]

\(xy+yz+zx+2xyz=1\)

\(\Leftrightarrow2xy+2yz+2zx+2x+2y+2z+2=xy+yz+zx+2x+2y+2z+3\)

\(\Leftrightarrow2\left(x+1\right)\left(y+1\right)\left(z+1\right)=\left(x+1\right)\left(y+1\right)+\left(y+1\right)\left(z+1\right)+\left(z+1\right)\left(x+1\right)\)

\(\Leftrightarrow\frac{1}{x+1}+\frac{1}{y+1}+\frac{1}{z+1}=2\)

\(\Rightarrow2\ge\frac{9}{x+y+z+3}\Rightarrow x+y+z\ge\frac{3}{2}\)

\(\Rightarrow x^2+y^2+z^2\ge\frac{1}{3}\left(x+y+z\right)^2\ge\frac{3}{4}\)

\(P^2\le3\left[3-\left(x^2+y^2+z^2\right)\right]\le3\left(3-\frac{3}{4}\right)=\frac{27}{4}\)

\(\Rightarrow P\le\frac{3\sqrt{3}}{2}\)

\(P_{max}=\frac{3\sqrt{3}}{2}\) khi \(x=y=z=\frac{1}{2}\)