Fix $\alpha \in (0,1)$ and $\psi\in C^{\infty}_{c}(\mathbb{R}\to \mathbb{R})$. For a smooth function $\phi\geq 0$ define the integral $$J_{\alpha}(\phi):=\int \frac{\psi}{\phi^{\alpha}}.$$ If $|\phi^{\prime}|$ is bounded below away from zero, then $J_{\alpha}$ exists for all $\alpha<1$. Indeed, one can simply integrate by parts to get $$J_{\alpha}=-\frac{1}{1-\alpha}\int \phi^{1-\alpha}\left(\frac{\psi}{\phi^{\prime}}\right)^{\prime}.$$
My question is about the case where $\phi^{\prime}$ changes sign finitely many times in the support of $\psi$. Using a smooth partition of unity, one may assume $\phi^{\prime}(x)=0$ at exactly one point $x_{0}$ in the support of $\psi$.
For simplicity, let us further assume that $|\phi^{\prime\prime}|$ is bounded below away from zero near $x_{0}$.
Obviously, we necessarily have $\alpha<1/2$. Just take $\phi$ to be a quadratic function.
I figured this out when $\phi^{\prime\prime}>C>0$: Write $$\phi(x)=\phi(x_{0})+(x-x_{0})^{2}\int_{0}^{1}(1-t) \phi^{\prime\prime}(x_{0}+t(x-x_{0}))dt\geq \frac{C}{2}(x-x_{0})^{2}$$ In this case we have $$|J_{\alpha}(\phi)|\leq \frac{2}{C}\int \frac{|\psi|}{|x-x_{0}|^{2\alpha}}$$ which exists for all $\alpha<1/2$.
What is left is the case $\phi^{\prime\prime}<c<0$.
Summary : Fix $\alpha \in (0,1/2)$ and $\psi\in C^{\infty}_{c}(\mathbb{R}\to \mathbb{R})$. Let $\phi\geq 0$ be a smooth function with a unique stationary point $x_{0}$ in the support of $\psi$. Assume that $\phi^{\prime\prime}<c<0$. Show that $J_{\alpha}(\phi)=\int \frac{\psi}{\phi^{\alpha}}$ is finite ?