Answer
Yes, $f(-c)$ is also a local maximum.
Work Step by Step
Step 1. Given $f(x)$ be an even function, we have $f(-x)=f(x)$.
Step 2. Assume the function has a local maximum at $x=c$; we have within an open interval $(a,b)$ ($a\lt c\lt b$), $f(c)\geq f(x)$
Step 3. The symmetry of the function indicates a reflection around the y-axis; we have for the interval of $(-b,-a)$ ($-b\lt -c\lt -a$), $f(-c)=f(c)\geq f(x)=f(-x)$ which means that $f(-c)$ is also a local maximum.
