Answer
$f(-c)=f(c)$ and at $x=-c$ there is also a local maximum.
Work Step by Step
Because f(x) is even, its graph is symmetric about the y-axis.
First, we know that $f(-c)=f(c).$
Next, the part of the graph around $P=(c,f(c)$ is such that $f(c)$ is the greatest value of $f(x)$ for some interval around x=c.
There exists a positive $h$ such that $x\in(c-h,c+h)\Rightarrow f(x)\leq f(c)$
Then, reflecting that part of the graph over the y-axis,
point P will be reflected to point $Q=(-c, f(c))$, and
the points that were below P (points for which $f(x)\leq f(c)$), will be reflected to points that are below Q.
This means that Q is a point of local maximum.
(If all points below P (in some local vicinity) have their y-coordinate not greater than f(c), then their reflections, points around Q, will also be below Q.)
