JEE Math Practice Question

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

1. A.$$f^{\prime \prime}(x)=0$$ for atleast two $$x$$ in $$(0,2)$$
2. B.$$f^{\prime}\left(\frac{3}{2}\right)+f^{\prime}\left(\frac{1}{2}\right)=1$$
3. C.$$f^{\prime \prime}(x)=0$$ for no $$x$$ in $$(0,1)$$
4. D.$$f^{\prime \prime}(x)=0$$ for exactly one $$x$$ in $$(0,1)$$