Let f be twice differentiable function satisfying f(1)=1, f(2)=4, f(3…

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 $x \in(1,3)$