Question

\(\left\{{}\begin{matrix}x^2+y^2=1\\x^3+y^3=1\end{matrix}\right.\)

Answer 1

Do \(x^2+y^2=1\Rightarrow\left\{{}\begin{matrix}\left|x\right|\le1\\\left|y\right|\le1\end{matrix}\right.\)

\(\Rightarrow\left\{{}\begin{matrix}x^3\le x^2\\y^3\le y^2\end{matrix}\right.\) \(\Rightarrow x^3+y^3\le x^2+y^2\)

Đẳng thức xảy ra khi và chỉ khi:

\(\left\{{}\begin{matrix}x^3=x^2\\y^3=y^2\\x^2+y^2=1\end{matrix}\right.\) \(\Leftrightarrow\left(x;y\right)=\left(1;0\right);\left(0;1\right)\)