Let two functions $f(x), g(x)$ be defined on an interval $E\subseteq\mathbb R$. On that interval, $f$ has $p$ isolated local maxima, and $g$ has $q$ isolated local maxima. By "isolated", I mean that if $x$ is an isolated maximum of $f$, then $\exists \epsilon>0$ such that $f(y)<f(x)$(strict inequality) whenever $|y-x|\leq \epsilon.$ Assume that all maxima of $f,g$ are isolated, so we are excluding the case where $f$ is a constant function on a subset of $E$.
Assume also that all the maxima of $f+g$ are isolated. $f+g$ has $r$ maxima. Given that $p,q\geq 1$, is it always true that $r\leq p+q$? Assume $f,g$ to be continuous or smooth or anything if we need it. Or is it true in some special cases(e.g. non-negative $f,g$)?
It is not true for $p=q=0$. Just take $f(x)=-x$ and $g(x)=x^3$ as a counterexample. However, I struggle to produce any counterexample for nontrivial values of $p,q$ (such as $2,3$).