MathematicsMedium63×since 2002Q3702Let g:R→Rg: \mathbf{R} \rightarrow \mathbf{R}g:R→R be a non constant twice differentiable function such that g′(12)=g′(32)\mathrm{g}^{\prime}\left(\frac{1}{2}\right)=\mathrm{g}^{\prime}\left(\frac{3}{2}\right)g′(21)=g′(23). If a real valued function fff is defined as f(x)=12[g(x)+g(2−x)]f(x)=\frac{1}{2}[g(x)+g(2-x)]f(x)=21[g(x)+g(2−x)], thenAf′′(x)=0f^{\prime \prime}(x)=0f′′(x)=0 for atleast two xxx in (0,2)(0,2)(0,2)Bf′(32)+f′(12)=1f^{\prime}\left(\frac{3}{2}\right)+f^{\prime}\left(\frac{1}{2}\right)=1f′(23)+f′(21)=1Cf′′(x)=0f^{\prime \prime}(x)=0f′′(x)=0 for no xxx in (0,1)(0,1)(0,1)Df′′(x)=0f^{\prime \prime}(x)=0f′′(x)=0 for exactly one xxx in (0,1)(0,1)(0,1)Check answerSkip