Determine all the differentiable functions $f : [0, 1] → R$ satisfying the conditions $f(0) = 0, f(1) = 1$ and $|f'(x)| ≤ 1/2$ for all $x \in [0, 1]$.
Solution 1:
$$|f(x)| = \left| \int_{0}^{x}f^{'}(t) \,dt \right| \le \int_0^{x} |f'(t)| \, dt \le \int_0^{x}\frac{1}{2} \,dt = \frac{x}{2}.$$ But $f(1) = 1$, so no such function.