Differentiation

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