For all twice differentiable functios

Let $f: \mathrm{R} \rightarrow \mathrm{R}$, with $f(0)=f(1)=0$ and $f^{\prime}(0)=0$

$\because f(x)$ is differentiable and continuous and

$f(0)=f(1)=0$

Then by Rolle's theorem, $f^{\prime}(c)=0, c \in(0,1)$

Now again

$\because f^{\prime}(c)=0, f^{\prime}(0)=0$

Then, again by Rolle's theorem,

Now again

$\because f^{\prime}(c)=0, f^{\prime}(0)=0$

Then, again by Rolle's theorem,

$f^{\prime \prime}(x)=0$ for some $x \in(0,1)$