On-line Math 21

4.5  Curve Sketching

Theorem 2 [The Second Derivative Test]. If f(x) has a critical point at c , then:

1. If f¢¢(c) > 0 , then f(x) has a relative minimum at c ,
2. If f¢¢(c) < 0 , then f(x) has a relative maximum at c , and
3. If f¢¢(c) = 0 , then you can't tell whether f(x) has a relative maximum, or minimum, or neither, at c .

Proof.

This actually reduces to the first derivative test, since f¢¢(c) > 0 means that f¢(x) is increasing at c , so that f¢(x) (which is 0 at c ) must go from negative to positive as you cross c , which describes a minimum. The idea for the second case is the same. For the last, you have to think about examples, to see that any of the possibilities might occur. Examples of this ``problem'' are:

1. f(x) = x⁴ . f¢(0) = 0 , and f¢¢(0) = 0 , but this is a local minimum,
2. f(x) = -x⁴ . f¢(0) = 0 , and f¢¢(0) = 0 , but this is a local maximum,
3. f(x) = x³ . f¢(0) = 0 , and f¢¢(0) = 0 , but this is neither a local minimum, no a local maximum.

Copyright (c) 2000 by David L. Johnson.