A calculus problem by Hummus a

Calculus Level 4

let \(F:[0,\pi ]\rightarrow \mathbb{R}\) be a smooth function with \(F(0)=F(\pi )=0\)

You're given that \(\displaystyle\int _{ 0 }^{ \pi }{ (F(x))^{ 2 } } dx=1\)

find the minimum value of \(\displaystyle\int _{ 0 }^{ \pi }{ \left (\dfrac { dF }{ dx } \right )^{ 2 } \, dx } \)

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