Let f be twice differentiable function satisfying $f(1)=1, f(2)=4, f(3)=9$, then

$f''(x)=2$ for all $x \in R$ $f'(x)=5=f''(x)$, for some $x \in[1,3]$ there exists at least one $x \in(1,3)$ such that $f''(x)=2$ none of these

there exists at least one $x \in(1,3)$ such that $f''(x)=2$

Let $g(x)=f(x)-x^2$. Then, $g(x)$ has at least three real roots in $[1,3]$ $\Rightarrow g'(x)$ has at least 2 real roots in $[1,3]$ $\Rightarrow g''(x)$ has at least one real root in $[1,3]$ $\Rightarrow f''(x)=2$ for at least one $x \in(1,3)$