Let \( f(x) \) be a non-constant twice differentiable function defi... VIDEO
Let \( f(x) \) be a non-constant twice differentiable function defined on \( (-\infty, \infty) \) such that \( f(x)=f(1-x) \) and \( f^{\prime}\left(\frac{1}{4}\right) \) \( =0 \). Then
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(1) \( f^{\prime}(x) \) vanishes at least twice on \( [0,1] \)
(2) \( f^{\prime}\left(\frac{1}{2}\right)=0 \)
(3) \( \int_{-1 / 2}^{1 / 2} f\left(x+\frac{1}{2}\right) \sin x d x=0 \)
(4) \( \int_{0}^{1 / 2} f(t) e^{\sin \pi t} d t=\int_{1 / 2}^{1} f(1-t) e^{\sin \pi t} d t \)
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