(O2) Second Derivative Test#
By the end of the lesson you will be able to:
classify all critical numbers using the second derivative test.
Critical Numbers#
First Derivative Test#
Second Derivative Test#
Example 1#
Example 2#
Example 3#
One Critical Number#
Example 4#
Finding Local Extrema#
Fermat’s Theorem
If \(f\) has a local maximum or minimum at \(x=c\), then either \(f'(c)=0\) or \(f'(c) \; DNE\).
Critical Number
The number \(x=c\) is a critical number of \(f\) provided either:
and \(c\) is in the domain of \(f\).
Classifying Critical Numbers#
Second Derivative Test for Local Extrema
Let \(x=c\) be a critical number of continuous function \(f\) such that \(f'(c)=0\) and \(f''\) is continuous near \(c\).
If \(f''(c)<0\), then \(f\) has a local maximum at \(x=c\).
If \(f''(c)>0\), then \(f\) has a local minimum at \(x=c\).
If \(f''(c)=0\), then the test is inconclusive.
Example 1#
Function \(f\) has critical numbers \(x=1\) and \(x=-2\). Use its second derivative given below to classify these critical numbers.
Example 2#
Find the local maximum and minimum values of:
Example 3#
Find the local maximum and minimum values of:
Example 4#
Find the local maximum and minimum values of:
Example 5#
Find the local maximum and minimum values of: