From now on, $f$ denotes an infinitely differentiable function from $[0,1]$ to $\mathbb{R}$. We assume that there exists a unique point $x_0 \in [0,1]$ where $f'$ vanishes. We also assume that $f''(x_0) > 0$.
For all $x \in [x_0, 1]$, we define
$$h(x) = \sqrt{|f(x) - f(x_0)|}$$
(a) Show that the function $h$ defines a bijection from $[x_0, 1]$ to $[0, h(1)]$.
(b) Show that the application $h$ is differentiable at $x_0$ from the right, and that $h'_+(x_0) = \sqrt{\frac{f''(x_0)}{2}}$.