Points on the graph of a function where the derivative is zero or the derivative does not exist are important to consider in many application problems of the derivative. The point (

*x, f(x)*) is called a

**critical point** of

*f(x)* if

*x* is in the domain of the function and either

*f′(x)* = 0 or

*f′(x)* does not exist. The geometric interpretation of what is taking place at a critical point is that the tangent line is either horizontal, vertical, or does not exist at that point on the curve.

**Example 1:** Find all critical points of .

Because *f(x)* is a polynomial function, its domain is all real numbers.

hence, the critical points of *f(x)* are (−2,−16), (0,0), and (2,−16).

**Example 2:** Find all critical points of *f(x)*= sin *x* + cos *x* on [0,2π].

The domain of *f(x)* is restricted to the closed interval [0,2π].

hence, the critical points of *f(x)* are and