If $f(x) \in C^1$ then number of maxima is finite in any interval. I am wondering if the following statement is true. 

If  $f(x) \in C^1$ (not-constant) then a number of maxima are finite in any finite interval.

Here the set $C^1$ means that on the domain of $f$ we have that
\begin{align}
\sup_{x \in dom(f(x))} |f(x)|<c_0<\infty,\\
\sup_{x \in dom(f(x))} |f^{'}(x)|<c_1<\infty,
\end{align}

By interval do you mean a finite interval. If not, the statement is not true. Counter example would be $sinc(x)$ on $(0,\infty)$
