Let $A$ be the set of twice continuously differentiable functions on $\,\left[0,1\right]\,$ and $$ B=\left\lbrace \;f\in A \mid \;f\left(0\right)=f\left(1\right)=0,\;\, f'\left(0\right)=2 \,\right\rbrace $$ Then what is $\;\displaystyle\min_{f\in B}\int_0^1\! \left(\,f''\left(x\right)\right)^2dx$?
I am not able to get anything out of the question except for taking a particular example of $\,f\left(x\right)=2x\left(1-x\right)$. Please can somebody tell the general thing?