Let $x_0$ be an extreme point of $f$ and $g$. Is $x_0$ an extreme point of $\max\{f,g\}$?

Question

Let $x_0$ be an local extreme point of $f$ and $g$. Is $x_0$ an local extreme point of $\max\{f,g\}$?

Let $x_0$ is the local maximum point of $f$ and $g$, it is OK! What may happen if $x_0$ is the local minimum point of $f$, and the local maximum point of $g$?

Answer 1

As written, $f,g$ are not required to be continuous. Let $$f(x)=\begin{cases}2&x=x_0\\0&\text{otherwise}\end{cases}$$ $$g(x)=\begin{cases}1&x<x_0\\0&x=x_0\\3&x>x_0\end{cases}$$ Then $f$ has a local maximum and $g$ a local minimum at $x_0$. However, $$\max\{f,g\}(x)=\begin{cases}1&x<x_0\\2&x=x_0\\3&x>x_0\end{cases}$$ does not have an extremum at $x_0$.