If $g$ and $f$ are periodic nonzero functions, and $g'f - f'g = 0$ at some point $x$, is it possible to have $g''f - f''g = 0$ at that point?

Question

Suppose I have two analytic periodic nonconstant function $f(x)$ and $g(x)$ with the same period, such that $0<g<f$ for all $x \in \mathbb{R}$,but their derivatives $f'$, $g'$ do have zeroes. Suppose that for at least one value of $x$ I have $$g'f - f'g = 0$$

Is it possible to have $g''f - f''g = 0$ also at that same value of $x$?

I'm thinking of "simple" functions like $f(x) = 5 + \sin x$ and $g(x) = 2 - \cos x$.

Thanks!