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Critical values of a cubic:


(Jump directly to the Test Problem)

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Sample problem:
cubic to use in finding the critical values

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

first derivative of the cubic

Setting this to 0 and factoring, we get
the factored first derivative for determining the critical numbers.

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:

Interval

To left of –3

Between –3 and 2

To the right of 2

Point in interval

 

 

 

Sign of f '(x) at point

 

 

 

Character of graph of f(x) at point

 

 

 

Sketch

 

 

 

First choose a point in each interval, near a boundary of the interval: