Answer
Yes, $g(-c)$ will be a local maximum.
Work Step by Step
Step 1. Given $g(x)$ is an odd function, we have $g(-x)=-g(x)$.
Step 2. If $g(x)$ has a local minimum value at $x=c$, we have for all $x$ in $(a, b), a\lt c\lt b$, $g(x)\geq g(c)$ and $-g(c)\geq -g(x)$
Step 3. At $x=-c$, consider symmetry of the function around the origin and the interval $(-b, -a), -b\lt -c\lt -a$, we have $g(-c)=-g(c)\geq -g(x)=g(-x)$ and this means that $g(-c)$ will be a local maximum.
