For the function, find all critical points or determine that no such points exist.
f (x) = sin x cos x
We find the derivative using the product rule.
This is never undefined. f '(x) is zero when
cos2 x = sin2 x,
cos x = ± sin x.
Think about the unit circle. sin and cos are equal at π/4, 3π/4, 5π/4, 7π/4, and anything you find by adding or subtracting 2π to any of these numbers. Look at a graph and see if this is believable:
Yup, it's believable. We have horizontal tangent lines everywhere we expected to. So our critical points occur at
where k is any integer.