Maximizing $x^2 y + y^2 z + z^2 x − x^2 z − y^2 x − z^2 y$ for $x$, $y$, $z$ between $0$ and $1$ (inclusive)

Question

Find the maximum value of the expression:-

$$x^2 y + y^2 z + z^2 x − x^2 z − y^2 x − z^2 y$$ when $0 \leq x \leq 1$, and $0 \leq y \leq 1$, and $0 \leq z \leq 1$.

Please note: This is a initial round contest math problem, which means two things:

1. Nothing beyond basic 10th grade algebra is required to solve it (only ingenuity)
2. I have not shown my approach simply because I don't know where to start and have made no progress (as is quite common with contest math problems)

Answer 1

Here are a couple of suggestions:

(i) if you rewrite this as a polynomial in $x$ with coefficients involving $y$ and $z$ (these will turn out just to be numbers after all) what do you find? [This is potentially motivated because the vertex of a parabola is easier to find than something involving a cubic expression, but also because this is a way of fitting something unfamiliar into a more familiar kind of pattern]

(ii) There are positive and negative terms here. What happens eg when two of the variables are equal? (or when the values of two variables are exchanged)

Do either of these approaches help you to find a way of rewriting the expression which might help you in your analysis?

Answer 2

Since the expression is cyclic, we can assume that $x=\max\{x,y,z\}$ and since $$\sum_{cyc}(x^2y-x^2z)=(x-y)(x-z)(y-z),$$ we can assume that $x\geq y\geq z$, otherwise $(x-y)(x-z)(y-z)\leq0.$

Now, by AM-GM $$(x-y)(x-z)(y-z)\leq\left(\frac{x-y+y-z}{2}\right)^2(x-z)=\frac{(x-z)^3}{4}\leq\frac{1}{4}.$$ The equality occurs for $(x,y,z)=\left(1,\frac{1}{2},0\right),$ which says that we got a maximal value.