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 points. The video includes a review of the test, outlines the steps to perform it, and demonstrates its application through examples involving transcendental and trigonometric functions. The tutorial concludes with a verification of results using graphs.

What is the primary purpose of the second derivative test?

• To find the roots of a function

• To calculate the integral of a function

• To determine the concavity of a function at critical points

• 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 has a relative minimum

• The function is undefined

• The function is concave downward

• The function has a relative maximum

What should be done if the second derivative equals zero at a critical point?

• Assume the function has a relative maximum

• Conclude the function is undefined

• Use the first derivative test

• Assume the function has a relative minimum

In the example with the function from zero to infinity, what was the critical number found?

• 0

• 1

• 2

• Infinity

What is the value of the second derivative at the critical number x = 1 in the first example?

• Undefined

• 1

• -1

• 0

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

• Tangent identity

• Sine double angle identity

• Pythagorean identity

• Cosine double angle identity

What are the critical numbers found in the trigonometric example?

• π/4, π/2, 3π/4

• π/6, π/3, π/2

• π/6, π/2, 5π/6

• π/3, π/2, 2π/3

What is the sign of the second derivative at x = π/2 in the trigonometric example?

• Negative

• Positive

• Zero

• Undefined

What is the relative extrema at x = π/2 in the trigonometric example?

• Relative maximum

• Relative minimum

• Absolute minimum

• Absolute maximum

How many relative maximums are identified in the trigonometric example?

• None

• Two

• One

• Three