Hi all,

Need some help with this simple? proof.

Let A be the set of all twice differentiable functions $\displaystyle f:[0,1] \rightarrow \mathbb{R}$ with the properties: $\displaystyle f(0) = 0$, $\displaystyle f(1) = 1$ and $\displaystyle f''\leq 0$.

Prove: If $\displaystyle f \in A$ and if

$\displaystyle \int_0^1 f(x) dx \leq \int_0^1 g(x) dx $

for all $\displaystyle g \in A$, then $\displaystyle f(x) = x$ with all $\displaystyle x \in [0,1]$.

Thanks.