Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 5.0 AIME, harder Find the answer

5. Find the maximum possible value of the expression $x y(x-y)$ , given that $x, y \in$ $[0 ; 1]$

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

Answer: 0.25.

Solution: Consider $x y(x-y)=-x y^{2}+x^{2} y$ as a quadratic function of $y$ . Its maximum is achieved at the vertex of the corresponding parabola $y_{0}=\frac{x}{2}$ . Clearly, it is located on the given segment. Substituting into the expression, we get $-x y_{0}^{2}+x^{2} y_{0}=\frac{x^{3}}{4}$ . Clearly, the maximum value is $\frac{1}{4}$ .

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.