Докажите, что: x^2(x-y)>=y^2(x-y), если x>=0 и y>=0.

Ответ:

\[x^{2}(x - y) \geq y^{2}(x - y);\ \ \ \ \ \ \]

\[x \geq 0;\ \ y \geq 0\]

\[(x - y)\left( x^{2} - y^{2} \right) \geq 0\]

\[(x - y)(x - y)(x + y) \geq 0\]

\[(x - y)^{2}(x + y) \geq 0\]

\[(x - y)^{2} > 0\ \ всегда;\ \ \]

\[x + y \geq 0 - из\ условия.\]