Understanding the Second Derivative Test

The video tutorial explains how to determine relative extrema of a function using the second derivative test. It begins with a review of the test, followed by an example problem to find critical numbers by setting the first derivative to zero. The second derivative is then evaluated at these critical numbers to determine concavity and identify relative maxima and minima. The tutorial concludes with calculating function values using a graphing calculator and verifying results graphically.

What is the first step in using the second derivative test to find relative extrema?

• Evaluate the second derivative at critical numbers

• Determine the critical numbers of the function

• Find the maximum and minimum values

• Graph the function

How do you find the critical numbers of a function?

• By graphing the function

• By evaluating the function at various points

• By finding where the first derivative is zero or undefined

• By setting the second derivative to zero

What does a positive second derivative at a critical number indicate about the function?

• The function has a relative maximum

• The function is concave down

• The function is concave up

• The function has no extrema

If the second derivative is negative at a critical number, what can be concluded?

• The function is concave up

• The function has a relative minimum

• The function has no extrema

• The function is concave down

What should be done if the second derivative test fails?

• Ignore the critical number

• Use the first derivative test

• Graph the function

• Recalculate the second derivative

How can the actual values of relative extrema be found after determining their locations?

• By solving the original function

• By using a graphing calculator

• By estimating from the graph

• By using the second derivative

What is the relative maximum value found in the example?

• 3

• -69

• 0

• 16 3/4

What is the relative minimum value found in the example?

• -69

• 3

• 16 3/4

• 0

How are relative maxima and minima sometimes expressed?

• As inequalities

• As decimals

• As ordered pairs

• As fractions

What does the point (-1/2, 16 3/4) represent in the context of the function?

• A relative minimum

• A point of inflection

• A relative maximum

• A critical point