Critical Point Definition : We say that x = c is a critical point of function f(x) if f (c) exists & if either of the given are true.

f ′ (c ) = 0        OR             f ′ (c ) doesn't exist

Note  as well that we require that f (c ) exists in order for x = c to in fact be a critical point. It is significant, & frequently overlooked, point.

The key point of this section is to work some instance finding critical points.  Thus, let's work some examples.

Example   Find out all the critical points for the function.

                      f ( x ) =6x⁵ + 33x⁴ - 30x³ + 100

Solution : First we need the derivative of the function to find the critical points & thus let's get that and notice that we'll factor out it as much as possible to make our life simple while we go to discover the critical points.

f ′ ( x ) =30 x⁴ + 132 x³ - 90 x²

(6 x² +5x² + 22 x -15)

( 6 x² (5x - 3) ( x + 5)

Now, our derivative is polynomial and therefore will exist everywhere.  So the only critical points will be those values of x that make the derivative zero.  Thus, we have to solve.

                                              6 x²(5x - 3) ( x + 5) = 0

Since this is the factored form of the derivative it's pretty simple to recognize the three critical points. They are,

                                      x = -5, x = 0, x = 3/5

Polynomials are generally fairly simple functions to find critical points for provided the degree doesn't get so large that we have trouble finding the roots of the derivative.

Most of the more "interesting" functions for finding critical points aren't polynomials however. Thus let's take a look at some functions that require a little more effort on our part.