General Contents

Detailed Contents

Index

Relative extrema of a cubic:

the cubic to use in finding the relative extrema
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The extrema or critical values are located where the derivative of f(x) is zero. We can test for a maximum or minimum by using the second derivative test. 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, displaying the x-coordinates of the extrema

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

Next we wish to use the Second Derivative Test to determine the sign of the second derivative of the graph of f(x) at these two values. The second derivative is

the second derivative of the cubic

Evaluate this at the critical points and determine their character as shown in the following table:

xc

–3

2

 

 

Sign of f ''(xc) at point

 

 

Character of graph of f(x) at point

 

 

Sketch

 

 

First evaluate the second derivative at the points: