\(\Leftrightarrow\left\{{}\begin{matrix}\dfrac{x^2+y^2+z^2}{25}=1\\\dfrac{x^3+y^3+z^3}{125}=1\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}\left(\dfrac{x}{5}\right)^2+\left(\dfrac{y}{5}\right)^2+\left(\dfrac{z}{5}\right)^2=1\\\left(\dfrac{x}{5}\right)^3+\left(\dfrac{y}{5}\right)^3+\left(\dfrac{z}{5}\right)^3=1\end{matrix}\right.\)
\(\Rightarrow\left(\dfrac{x}{5}\right)^2+\left(\dfrac{y}{5}\right)^2+\left(\dfrac{z}{5}\right)^2=\left(\dfrac{x}{5}\right)^3+\left(\dfrac{y}{5}\right)^3+\left(\dfrac{z}{5}\right)^3\)
Do \(\left(\dfrac{x}{5}\right)^2+\left(\dfrac{y}{5}\right)^2+\left(\dfrac{z}{5}\right)^2=1\Rightarrow\left\{{}\begin{matrix}\left|\dfrac{x}{5}\right|\le1\\\left|\dfrac{y}{5}\right|\le1\\\left|\dfrac{z}{5}\right|\le1\end{matrix}\right.\)
\(\Rightarrow\left\{{}\begin{matrix}\left(\dfrac{x}{5}\right)^2\ge\left(\dfrac{x}{5}\right)^3\\\left(\dfrac{y}{5}\right)^2\ge\left(\dfrac{y}{5}\right)^3\\\left(\dfrac{z}{5}\right)^2\ge\left(\dfrac{z}{5}\right)^3\end{matrix}\right.\) \(\Rightarrow\left(\dfrac{x}{5}\right)^2+\left(\dfrac{y}{5}\right)^2+\left(\dfrac{z}{5}\right)^2\ge\left(\dfrac{x}{5}\right)^3+\left(\dfrac{y}{5}\right)^3+\left(\dfrac{z}{5}\right)^3\)
Dấu "=" xảy ra khi và chỉ khi:
\(\left\{{}\begin{matrix}\left(\dfrac{x}{5}\right)^2=\left(\dfrac{x}{5}\right)^3\\\left(\dfrac{y}{5}\right)^2=\left(\dfrac{y}{5}\right)^3\\\left(\dfrac{z}{5}\right)^2=\left(\dfrac{z}{5}\right)^3\\\left(\dfrac{x}{5}\right)^2+\left(\dfrac{y}{5}\right)^2+\left(\dfrac{z}{5}\right)^2=1\end{matrix}\right.\)
\(\Rightarrow\left(x;y;z\right)=\left(5;0;0\right);\left(0;5;0\right);\left(0;0;5\right)\)
