Understanding Absolute Extrema

The video tutorial explains how to determine the absolute extrema of a continuous function over a closed interval. It covers finding critical numbers by setting the derivative to zero, evaluating the function at endpoints and critical numbers, and verifying results graphically. The process involves factoring the derivative, checking critical numbers within the interval, and calculating function values to identify the absolute minimum and maximum.

What is the first step in determining the absolute extrema of a function over a closed interval?

• Ensure the function is continuous over the interval.

• Find the second derivative of the function.

• Check if the function is differentiable.

• Determine the function's domain.

How do you find the critical numbers of a function?

• By evaluating the function at the endpoints.

• By finding where the derivative is zero or undefined.

• By checking the continuity of the function.

• By setting the second derivative to zero.

What is the greatest common factor used in factoring the derivative 4x^3 - 64x?

• 8x

• 16x

• 4x

• 2x

Which critical number is not evaluated because it is outside the interval from -3 to 9?

• x = 9

• x = -4

• x = 0

• x = 4

What is the function value at x = 4?

• 3978

• -247

• 9

• -198

What is the absolute maximum value of the function over the interval?

• -198

• -247

• 3978

• 9

At which x-value does the absolute minimum occur?

• x = 9

• x = 0

• x = 4

• x = -3

What is the purpose of graphically verifying the function's extrema?

• To find new critical points.

• To confirm the calculated extrema are correct.

• To calculate the derivative again.

• To determine the function's continuity.

Which point on the graph represents the absolute maximum?

• The point where y = -247

• The point where y = 9

• The point where y = 3978

• The point where y = -198

What is the y-value of the lowest point on the graph over the interval?

• 9

• -198

• -247

• 3978