Find the critical values of a cubic:

The extrema or critical values are located where the derivative of f(x) is zero. We can use the Power, Scalar Multiple, and Sum Rules to get

Setting this to 0 and factoring, we get

Since f ' is defined for all values of x, the critical numbers are -3 and +2.

Next we wish to use the First Derivative Test to determine the slope of the graph of f(x) in the 3 intervals defined by these two values as shown in the following table:

From the table we conclude that the function f(x) must have a maximum at x = -3 and a minimum at x = +2.

Test Problem: Determine the location and nature of the critical points for

Which of the following are the x-coordinates of the critical points?
a) 1 and -4
b) -1 and 4
c) -1 and -4
d) 1 and 4

Or you can return to the beginning of sample problem.