Let $I =[a,b]$, and let $\theta$ denote a smooth function $I \to (0, \pi)$. Consider the functional
$$ J(\theta) = \int_{a}^{b} \frac{\left(F(s,\theta(s))^{2}+G(s,\theta'(s))^{2}\right)^{2}}{F(s,\theta(s))^{2}}\, ds, $$
where both $F$ and $G$ are continuous (in fact, smooth) functions.
Question. Suppose that $G(s,\theta'(s)) =0$ whenever $F(s,\theta(s)) =0$ and that the set of points where $F(s,\theta(s))=0$ is nowhere dense in $I$. Can we then conclude that $J$ has a positive lower bound?
Intuitively, it seems reasonable to expect the answer to be affirmative, but I am lacking a proof.
The functional $J$ is interesting to me because it measures the bending energy of an infinitely narrow developable strip in $\mathbb{R}^{3}$.