Understanding the Second Derivative Test

This video tutorial provides a detailed explanation of the second derivative test, a method used to find relative extrema of functions by analyzing the sign of the second derivative at critical numbers. The video includes a review of the test, outlines the steps to perform it, and provides two examples: one using a simple function and another involving trigonometric functions. The tutorial emphasizes the importance of determining concavity to identify relative minima and maxima, and it concludes with a verification of results using graphs.

What is the primary purpose of the second derivative test?

• To solve differential equations

• To calculate the integral of a function

• To determine the concavity and relative extrema of a function

• To find the absolute maximum of a function

If the second derivative at a critical number is positive, what can be inferred about the function?

• The function is concave down at that point

• The function has a relative maximum at that point

• The function is concave up at that point

• The function has an inflection point at that point

What should be done if the second derivative test fails?

• Ignore the critical number

• Recalculate the second derivative

• Use the integral test

• Use the first derivative test

In the example with the function 1 - 1/X, what is the critical number found?

• X = -1

• X = 0

• X = 1

• X = 2

What is the value of the second derivative at the critical number X = 1 for the function 1 - 1/X?

• 0

• 1

• -1

• 2

In the trigonometric example, which identity is used to simplify the expression?

• Cosine double angle identity

• Sine double angle identity

• Tangent double angle identity

• Pythagorean identity

What are the critical numbers found in the trigonometric example?

• X = PI/6, X = 5PI/6, X = PI/2

• X = PI/2, X = PI

• X = PI/3, X = 2PI/3

• X = PI/4, X = 3PI/4

What is the sign of the second derivative at X = PI/6 in the trigonometric example?

• Zero

• Undefined

• Negative

• Positive

What is the relative extrema at X = PI/2 in the trigonometric example?

• Relative minimum

• No extrema

• Relative maximum

• Inflection point

How can the results of the second derivative test be verified?

• By using a graph

• By recalculating the derivatives

• By checking the integral

• By using a calculator