Let $\mathrm{f}(\mathrm{x})$ be a non-constant twice differentiable function defined on $(-\infty, \infty)$ such that $\mathrm{f}(\mathrm{x})=\mathrm{f}(1-\mathrm{x})$ and $\mathrm{f}^{\prime}\left(\frac{1}{4}\right)=0$. Then
Select ALL correct options:
A
$\mathrm{f}^{\prime \prime}(\mathrm{x})$ vanishes at least twice on $[0,1]$
B
$\mathrm{f}^{\prime}\left(\frac{1}{2}\right)=0$
C
$\int_{-1 / 2}^{1 / 2} f\left(x+\frac{1}{2}\right) \sin x d x=0$
D
$\int_0^{1 / 2} f(t) e^{\sin \pi t} d t=\int_{1 / 2}^1 f(1-t) e^{\sin \pi t} d t$
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