For all twice differentiable functions

$f(0)=f(1)=f^{\prime}(0)=0$

Apply Rolles theorem on $\mathrm{y}=f(\mathrm{x})$ in $\mathrm{x} \in[0,1]$

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

$\Rightarrow f^{\prime}(\alpha)=0$ where $\alpha \in(0,1)$

Now apply Rolles theorem on $\mathrm{y}=f^{\prime}(\mathrm{x})$

in $x \in[0, \alpha]$

$f^{\prime}(0)=f^{\prime}(\alpha)=0$ and $f^{\prime}(x)$ is continuous and differentiable

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

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