Understanding Critical Numbers and Relative Extrema

The video tutorial explains how to find critical numbers of a function and determine if they correspond to relative maxima or minima. It covers the application of the product and chain rules to find the first derivative, simplifies the derivative, and uses the first derivative test to identify relative extrema. The tutorial concludes with calculating the relative maximum value and verifying it using a graph.

What is the purpose of finding the critical numbers of a function?

• To calculate the integral of the function

• To determine where the function is undefined

• To identify the domain of the function

• To find the points where the function has a relative maximum or minimum

Which rule is applied to find the first derivative of a product of two functions?

• Quotient Rule

• Product Rule

• Chain Rule

• Power Rule

What is the derivative of e raised to the power of -6x?

• e to the power of -6x times 6

• e to the power of 6x times 6

• e to the power of -6x times -6

• e to the power of 6x times -6

What is the critical number found in the function discussed?

• 3/4

• 2/3

• 1/2

• 1/3

What does the first derivative test help determine?

• The absolute maximum of the function

• The relative maximum or minimum at a critical number

• The concavity of the function

• The points of inflection

Which interval is used to test the sign of the first derivative to the left of the critical number?

• (2/3, ∞)

• (-∞, 0)

• (-∞, 2/3)

• (-8, 2/3)

What is the sign of the first derivative at x = 0?

• Positive

• Negative

• Zero

• Undefined

What is the relative maximum value of the function?

• 1 divided by 3e to the fourth

• 1 divided by e to the fourth

• 3e to the fourth

• 3 divided by e to the fourth

How is the relative maximum expressed as an ordered pair?

• (2/3, 3 divided by e to the fourth)

• (1/3, 1 divided by 3e to the fourth)

• (2/3, 1 divided by 3e to the fourth)

• (1/3, 3 divided by e to the fourth)

What does the graph of the function verify?

• The function is always decreasing

• The function has no critical points

• The location and value of the relative maximum

• The function is always increasing