For the function, use the second derivative test (if possible) to determine if each critical point is a minimum, maximum, or neither. If the second derivative test can't be used, say so.
The derivative is
f '(x) = -sin x
and the second derivative is
f "(x) = -cos x.
On the interval specified, f ' is zero at
x = -π, 0, π
and never undefined, so these are all the critical points we need to worry about.Now we evaluate the second derivative at each critical point and see what we find.
f "(-π) = -cos(-π) = 1 > 0
so f is concave up and has a minimum at x = -π.
f "(0) = -cos 0 = -π < 0
so f is concave down and has a maximum at x = 0.
f "(π) = -cos(π) = 1 > 0
so f is concave up and has a minimum at x = π.