Let f(x) be a twice differentiable function for all real values of x and satisfies $\mathrm{f}\left(1\right)=1, \mathrm{f}\left(2\right)=4, \mathrm{f}\left(3\right)=9.$ Then which of the following is definitely true? 

$\mathrm{Let} \mathrm{g}\left(\mathrm{x}\right)=\mathrm{f}\left(\mathrm{x}\right)-{\mathrm{x}}^2. \mathrm{We} \mathrm{have} \mathrm{g}\left(1\right)=0, \mathrm{g}\left(2\right)=0, \mathrm{g}\left(3\right)=0 \left[ \mathrm{f}\left(1\right)=1, \mathrm{f}\left(2\right)=4, \mathrm{f}\left(3\right)=9\right].$

From Rolle ' s theorem on $\mathrm{g}\left(\mathrm{x}\right).\mathrm{g}\text{'}\left(\mathrm{x}\right)=0$ for at least one $\mathrm{x}\in \left(1,2\right)$

$\mathrm{Let} \mathrm{g}\text{'}\left({\mathrm{c}}_1\right)=0 \mathrm{where} {\mathrm{c}}_1\in \left(1,2\right)$

Similarly, g(x) = 0 for at least one  $\mathrm{x}\in \left(2,3\right). \mathrm{Let} \mathrm{g}\text{'}\left({\mathrm{c}}_2\right)=0$ where ${\mathrm{c}}_2\in \left(2,3\right)$. Therefore

    ,$\mathrm{g}\text{'}\left({\mathrm{c}}_1\right)=\mathrm{g}\text{'}\left({\mathrm{c}}_2\right)=0$

By Rolle's theorem, at least one $\mathrm{x}\in \left({\mathrm{c}}_1,{\mathrm{c}}_2\right)$ such that $\mathrm{g}\text{'}\text{'}\left(\mathrm{x}\right)=0 \mathrm{or} \mathrm{f}\text{'}\text{'}\left(\mathrm{x}\right)=2 \mathrm{for} \mathrm{some} \mathrm{x}\in \left(1,3\right).$