Answer
$g$ has a local maximum value at $c$.
Work Step by Step
Suppose $f$ has a local minimum value at $c$.
Then there is some number $\epsilon \gt 0$ such that $f(c) \leq f(x)$ for all values of $x$ where $c-\epsilon \lt x \lt c+\epsilon$. That is, $f(c) \leq f(x)$ for all values of $x$ that are near $c$.
Let $g(x) = -f(x)$
Choose any number $x$ that is near $c$. That is, choose any number $x$ such that $c-\epsilon \lt x \lt c+\epsilon$.
Then:
$f(c) \leq f(x)$
$-f(c) \geq -f(x)$
$g(c) \geq g(x)$
Then, $g$ has a local maximum value at $c$.
