RO-NMO 2015 Problem 123

Let be the set $\mathcal{C} =\left\{ f:[0,1]\longrightarrow\mathbb{R}\left| \exists f''\bigg|_{[0,1]} \right. \exists x_1,x_2\in [0,1] x_1\neq x_2\wedge \left( f\left( x_1 \right) = f\left( x_2 \right) =0\vee f\left( x_1 \right) = f'\left( x_1 \right) = 0\right) \wedge f''<1 \right\} ,$

and $f^*\in\mathcal{C}$ such that $\int_0^1\left| f^*(x) \right| dx =\sup_{f\in\mathcal{C}} \int_0^1\left| f(x) \right| dx .$

Find $\int_0^1\left| f^*(x) \right| dx$ and describe $f^*.$