Understanding Absolute Extrema

The video tutorial explains how to determine absolute extrema on a closed interval, highlighting the differences between absolute and relative extrema. It covers evaluating function values at critical numbers and endpoints, using graphical analysis, and calculating extrema with a calculator. The tutorial concludes with a verification of results and emphasizes the importance of checking both critical numbers and endpoints.

What is the main difference between absolute extrema and relative extrema?

• Relative extrema are only at endpoints.

• Absolute extrema are only at critical points.

• Relative extrema are always larger.

• Absolute extrema can occur at endpoints.

To find the absolute extrema on a closed interval, what must be evaluated?

• Neither critical numbers nor endpoints.

• Both critical numbers and endpoints.

• Only the critical numbers.

• Only the endpoints of the interval.

What is the first step in determining the critical numbers of a function?

• Evaluate the function at endpoints.

• Find the second derivative.

• Set the first derivative equal to zero.

• Graph the function.

Which of the following is a critical number for the function x^3 - 3x^2?

• x = 0

• x = 4

• x = 1

• x = 2

When using a calculator to evaluate function values, what is the first step?

• Find the second derivative.

• Solve for x.

• Enter the function into Y1.

• Graph the function.

What is the absolute maximum value of the function on the interval from -1 to 4?

• 5 at x = 6

• -6.75 at x = 3

• 1.25 at x = -1

• 0 at x = 0

What is the absolute minimum value of the function on the interval from -1 to 4?

• 5 at x = 6

• 0 at x = 0

• -6.75 at x = 3

• 1.25 at x = -1

Why is it important to verify absolute extrema graphically?

• To confirm the calculated values visually.

• To determine the function's domain.

• To find new critical points.

• To ensure the function is continuous.

What does the graphical verification show about the absolute maximum?

• It is the same as the relative maximum.

• It occurs at a critical number.

• It occurs at an endpoint.

• It is not present on the graph.

What does the graphical verification show about the absolute minimum?

• It is not present on the graph.

• It occurs at an endpoint.

• It occurs at a critical number.

• It is the same as the relative minimum.