Calculus of a Single Variable, Chapter 5, 5.8, Section 5.8, Problem 37

This function is infinitely differentiable on entire RR. The necessary condition of extremum for such a function is f'(x) = 0.
To find the derivative of this function we need the product rule and the derivatives of sine, cosine, hyperbolic sine and hyperbolic cosine. We know them:)
So f'(x) = cosx sinhx + sinx coshx + sinx coshx - cosx sinhx = 2 sinx coshx.
The function coshx is always positive, hence f'(x) = 0 at those points where sinx = 0. They are k pi for integer k, and three of them are in the given interval: -pi, 0 and pi.
Moreover, f'(x) has the same sign as sinx, so it is positive from -4 to -pi, negative from -pi to 0, positive from 0 to pi and negative again from pi to 4. Function f(x) increases and decreases accordingly, therefore it has local minima at x=-4, x=0 and x=4, and local maxima at x=-pi and x=pi.