(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:

\[ f'(c)=0 \qquad \text{or} \qquad f'(c) \;\text{ DNE} \]

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.

\[ f''(x)= 4x^3-15x^2+2x+21 \]

Example 2#

Find the local maximum and minimum values of:

\[ f(x)= \dfrac{1}{3}x^3-2x^2+3x+1 \]

Example 3#

Find the local maximum and minimum values of:

\[ f(x)= x^4-4x^3 \]

Example 4#

Find the local maximum and minimum values of:

\[ f(x)= \dfrac{2x^2}{x^2-1} \]

Example 5#

Find the local maximum and minimum values of:

\[ f(x)= \ln(1-x^2) \]