This is exercise 2.2(b) in Brian Hall's Quantum Theory for Mathematicians.
Given a continuously differentiable potential energy function $V:{\bf R}\rightarrow{\bf R}$.
Suppose that $V(x)<E_0$ for $x_0\leq x<x_1$, with $V(x_1)=E_0$ and $V'(x_1)=0$.
Show that,
$$\int_{x_0}^{x_1}\sqrt{\frac{m}{2(E_0-V(y))}}dy=\infty.$$
I spent a while trying to unsuccessfully prove this, and began to suspect it was in fact not true.
Here is the counterexample I came up with:
Let $E_0=1$ and consider the following potential energy function on the interval $[0,1]$,
$$V(x)=-(x-1)^\frac{4}{3}+1.$$
Then $V(x) < E_0$ on $[0,1)$ with $V(x_1)=E_0$, and $V'(x_1)=-\frac{4}{3}(x_1-1)^\frac{1}{3}=0$.
Observe that the integral is finite,
$$\int_0^1\sqrt{\frac{m}{2(1+(y-1)^\frac{4}{3}-1)}}dy=\sqrt{\frac{m}{2}}\int_0^1\frac{1}{(y-1)^\frac{2}{3}}dy=\sqrt{\frac{m}{2}}3(y-1)^\frac{1}{3}\Biggr\rvert_{0}^1=3\sqrt{\frac{m}{2}}.$$
Am I missing something here? Is the exercise really incorrect as stated?
I'm aware of the answer given here, however it's too handwavy for me to follow.
Edit:
$\color{blue}{\text{Since my counterexample is not smooth, I suspect that maybe I need to assume that the}}$ $\color{blue}{\text{potential $V$ is smooth for this to be true.}}$
If I assume that $V$ is smooth, could anyone give me some help on how to use this fact to solve this?