Can anybody give me a constructive approach of how to tackle this kind of problems?
2026-03-31 10:01:36.1774951296
optimization problem entrance
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take a derivative to find a stationary point: $(x(1-x))' = 1-2x=0$. Thus, at $x^*=0.5$ the function has the maximum $0.25$. It is maximum since the second derivative is negative. For any $0 \lt \beta \lt 0.5,$ we'll have $x^* = \beta$.