Understanding Absolute Extrema

The video tutorial explains how to find the absolute extrema of a continuous function over a closed interval. It begins by identifying critical numbers where the derivative is zero or undefined. The derivative is solved to find these critical points, and the function is evaluated at these points and the interval's endpoints. The function values are compared to determine the absolute maximum and minimum. Finally, the results are verified graphically to ensure accuracy.

What ensures the existence of absolute extrema for a continuous function over a closed interval?

• The function is differentiable.

• The function is continuous.

• The function is decreasing.

• The function is increasing.

What is the derivative of the term 24x^2 in the function?

• 24x

• 48x

• 12x

• 36x

What type of function is the derivative in this problem?

• Linear

• Exponential

• Quadratic

• Cubic

What are the critical numbers found by solving the derivative equation?

• x = -1 and x = 9

• x = 0 and x = 1

• x = -9 and x = 1

• x = -9 and x = 2

Which critical number is within the given interval?

• x = 2

• x = 1

• x = 0

• x = -9

At which x-value does the function have its absolute minimum?

• x = 2

• x = 1

• x = 0

• x = -9

What is the absolute maximum value of the function?

• 54

• 7

• 975

• -25

What is the function value at x = 2?

• 7

• -25

• 975

• 0

How is the absolute maximum verified graphically?

• By finding the intersection with the x-axis.

• By identifying the highest point on the graph.

• By calculating the area under the curve.

• By checking the slope of the tangent.

What does the graph confirm about the function's absolute minimum?

• It occurs at x = 2.

• It occurs at x = 1.

• It occurs at x = 0.

• It occurs at x = -9.