Consider a degree-$5$ polynomial function $f:(-\infty, \infty) \rightarrow(-\infty, \infty)$. If $f$ exhibits at least four local maxima, which of the following is necessarily true? (Note: A local maximum is a point where the function value is the maximum in a sufficiently small neighbourhood.)
- $f(x)>0, x \in(-\infty, \infty)$
- $f(50)<0$
- The seventh derivative of $f(x)$ is negative for some $x \in[0,100]$
- $f$ has exactly $4$ local maxima
- None of the above