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for $x \in \mathbb{R}$, consider $f(x) = x(1-x)$, using traditional methods of finding global extremas, we can show that the derivative has a unique zero at $x= \frac12$ and $f''(\frac12) < 0$, thus $x(1-x) \leq \frac14 = f(\frac12)$

is there a more elegant way ?

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