Let g: R → R be a non constant twice differentiable function such that g^′(1/2)= g^′(3/2). If a real valued function f is defined as…

Let $\displaystyle \mathrm{g}: \mathrm{R} \rightarrow \mathrm{R}$ be a non constant twice differentiable function such that $\displaystyle \mathrm{g}^{\prime}\left(\frac{1}{2}\right)=\mathrm{g}^{\prime}\left(\frac{3}{2}\right)$. If a real valued function $\displaystyle f$ is defined as $\displaystyle f(x)=\frac{1}{2}[g(x)+g(2-x)]$, then