Official solution
\[f^{\prime}(x)=\frac{g^{\prime}(x)-g^{\prime}(2-x)}{2}, f^{\prime}\left(\frac{3}{2}\right)=\frac{g^{\prime}\left(\frac{3}{2}\right)-g^{\prime}\left(\frac{1}{2}\right)}{2}=0\]
Also \[\mathrm{f}^{\prime}\left(\frac{1}{2}\right)=\frac{\mathrm{g}^{\prime}\left(\frac{1}{2}\right)-\mathrm{g}^{\prime}\left(\frac{3}{2}\right)}{2}=0, \mathrm{f}^{\prime}\left(\frac{1}{2}\right)=0\]
\[\Rightarrow \mathrm{f}^{\prime}\left(\frac{3}{2}\right)=\mathrm{f}^{\prime}\left(\frac{1}{2}\right)=0\]
\[\Rightarrow \operatorname{rootsin}\left(\frac{1}{2}, 1\right)\] and \[\left(1, \frac{3}{2}\right)\]
\[\Rightarrow \mathrm{f}^{\prime \prime}(\mathrm{x})\] is zero at least twice in \[\left(\frac{1}{2}, \frac{3}{2}\right)\]