Question

Tìm tất cả các số thực x,y,z biết

x\(\sqrt{1-y^2}\)+y\(\sqrt{2-z^2}\)+z\(\sqrt{3-x^2}\)=3

Answer 1

Áp dụng BĐT \(ab\le\dfrac{a^2+b^2}{2}\)

\(x\sqrt{1-y^2}+y\sqrt{2-z^2}+z\sqrt{3-x^2}\le\dfrac{x^2+1-y^2}{2}+\dfrac{y^2+2-z^2}{2}+\dfrac{z^2+3-x^2}{2}=\dfrac{6}{2}=3\)

Dấu "=" xảy ra nên:

\(\left\{{}\begin{matrix}x=\sqrt{1-y^2}\\y=\sqrt{2-z^2}\\z=\sqrt{3-x^2}\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}x^2=1-y^2\\y^2=2-z^2\\z^2=3-x^2\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}x^2=1-\left(2-z^2\right)=z^2-1\\z^2=3-x^2\end{matrix}\right.\)

\(\Rightarrow x^2=3-x^2-1=2-x^2\Rightarrow x^2=1\Rightarrow x=1\Rightarrow y=0\Rightarrow z=\sqrt{2}\)

Vậy \(\left\{{}\begin{matrix}x=1\\y=0\\z=\sqrt{2}\end{matrix}\right.\)