Let $f\left(x\right)$ be a non-constant twice differentiable function defined on $\left(-\infty ,\infty \right)$ such that $f\left(x\right)=f\left(1-x\right)and{f}^{prime}\left(\frac{1}{4}\right)=0.$ Then (a)$f^{prime}\left(x\right)$ vanishes at least twice on $\left[0,1\right]$ (b)$f^{prime}\left(\frac{1}{2}\right)=0$ (c)${\int }_{-\frac{1}{2}}^{\frac{1}{2}}f\left(x+\frac{1}{2}\right)\sin xdx=0$ (d)${\int }_{-\frac{1}{2}}^{\frac{1}{2}}f\left(t\right)e^{\sin \pi t}dt={\int }_{\frac{1}{2}}^{1}f\left(1-t\right)e^{\sin \pi t}dt$

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