Understanding Absolute Extrema on Open Intervals

The video tutorial explains how to determine absolute extrema on open intervals, contrasting it with closed intervals. It highlights the importance of analyzing critical numbers and end behavior, using examples to illustrate when a function has absolute minima or maxima. The process of finding critical numbers through derivatives is detailed, followed by evaluating function values at these points. Finally, graph analysis is used to confirm the presence of absolute extrema.

What is the main difference between determining absolute extrema on open and closed intervals?

• Testing function values at endpoints is only necessary for open intervals.

• Closed intervals require testing at endpoints, while open intervals do not.

• Open intervals require testing at endpoints, while closed intervals do not.

• There is no difference between open and closed intervals in this context.

In the example with the function on the left, why is there no absolute maximum?

• The function decreases without bound.

• The function has no critical numbers.

• The function increases without bound.

• The function is constant.

For the function on the right, what makes the relative maximum also an absolute maximum?

• The function decreases without bound.

• The function is constant.

• The function has no minimum.

• There are no function values greater than the relative maximum.

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

• Graphing the function.

• Evaluating the function at endpoints.

• Setting the first derivative equal to zero.

• Finding the second derivative.

Which of the following is a critical number for the given function?

• x = 1

• x = 3

• x = -2

• x = 5

What tool is used to evaluate the function at critical numbers in the example?

• Integral calculator

• Derivative calculator

• Graphing calculator

• Statistical software

Why is -20 considered an absolute minimum in the example?

• It is the greatest function value on the interval.

• It is the least function value on the interval.

• It is a relative maximum.

• It is the only critical number.

What is the reason for the absence of an absolute maximum in the example?

• The function decreases to the left and right.

• The function has no critical numbers.

• The function increases to the left and right.

• The function is constant.

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What is the absolute minimum value found in the example?

• -2

• -20

• 0

• -4

At which x-values does the absolute minimum occur in the example?

• x = 0 and x = 1

• x = -2 and x = 2

• x = 0 and x = 2

• x = -1 and x = 1