<s>[INST] If $a$ and $b$ are non-negative real numbers, prove that $ab(a+b) \leq a^2+b^2$. 
If $a$ and $b$ are non-negative real numbers, prove that $ab(a+b) \leq a^2+b^2$.

Is is a geometric mean? How to prove it?
 [/INST] Note that if $a+b\le 2$, then $ab(a+b)\le 2ab$. And we have $2ab\le a^2+b^2$, since $(a-b)^2\ge 0$. So the desired inequality does hold if $a+b\le 2$. 
If the condition $a+b\le 2$ is violated, then the inequality can fail. For let $a=b=k$. Then $ab(a+b)=2k^3$ and $a^2+b^2=2k^2$. If $k\gt 1$ then $2k^3\gt 2k^2$. 
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