For the function, find all critical points or determine that no such points exist.

*f* (*x*) = sin *x* cos *x*

Answer

*f* (*x*) = sin *x *cos *x*

We find the derivative using the product rule.

This is never undefined. *f *'(*x*) is zero when

cos^{2 }*x* = sin^{2 }*x*,

or when

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.