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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
(a) \( f^{\prime}(x) \) vanishes at least twice on \( [0,1] \)
(b) \( f(1 / 2)=0 \)
(c) \( \int_{-1 / 2}^{1 / 2} f(x+1 / 2) \sin x d x=0 \)
(d) \( \int_{n}^{1 / 2} f(t) e^{\sin \pi t} d t=\int_{1 / 2}^{1} f(1-t) e^{\sin \pi t} d t \)
\( \mathrm{P} \)
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