Answer
Proof given below.
Work Step by Step
$f''$ is contiuous $\Rightarrow f'$ is differentiable on $(a,b) $(and continuous on $[a,b].$)
$\Rightarrow f$ is differentiable on $(a,b) $(and continuous on $[a,b].$)
There exist $a \leq c_{1} \ \lt c_{2} \ \lt c_{3}\leq b$
such that $f(c_{1})=f(c_{2})=f(c_{3})=0$.
f satisfies conditions of Rolle's theorem on $\ [c_{1},c_{2}]\ \Rightarrow$ there is a $d_{1}\in(c_{1},c_{2})$ such that $f'(d_{1})=0$
f satisfies conditions of Rolle's theorem on $\ [c_{2},c_{3}]\ \Rightarrow$ there is a $d_{2}\in(c_{2},c_{3})$ such that $f'(d_{2})=0$
$f'$ satisfies conditions of Rolle's theorem on $\ [d_{1},d_{2}]\ \Rightarrow$ there is an $e\in(d_{1},d_{2})$ such that $f''(e)=0$,
which proves the statement.
---
To generalize,
if $f^{(n)}$ is continuous on $[a,b]$ and f has $n+1$ zeros in the interval, then $f^{(n)}$ has at least one zero in $(a,b).$