How find the maximum of the $x^3_{1}+x^3_{2}+x^3_{3}-x_{1}x_{2}x_{3}$ Let $$0\le x_{i}\le i,\, i=1,2,3$$ be real numbers. Find the maximum of the expression
$$x^3_{1}+x^3_{2}+x^3_{3}-x_{1}x_{2}x_{3}$$

My idea: I guess
$$x^3_{1}+x^3_{2}+x^3_{3}-x_{1}x_{2}x_{3}\le 0^3+2^3+3^3-0\cdot2\cdot 3=35$$ 
But I can't prove it. Can you help ?
This problem is a special case of:


Hint:
As the objective function is convex and continuous, and the domain of interest is compact and convex, so we must have maximum when each $x_i \in \{0, i\}$.  This cuts down the problem (even the general $n$ variable case) to checking a few extreme cases, many of which are trivial...
