Question 1

Which of the following best describes the absolute maximum of a function $$f(x) on a $$closed interval $$[a, b]$$?

Accepted Answer

It is the value $$f(c)$$ where $$f(c) \u003e f(x)$$ for all $$x in $$the domain of $$f.$$

Question 2

Given $$f(x) = 9 - x^2$ on $$[-4, 4]$$, what is the absolute maximum value of f(x$$)$$ on this interval?

Accepted Answer

$$9$$

Question 3

For the function $$f(x) = 9 - x^2$ on $$[-4, 4]$$, what is the absolute minimum value?

Accepted Answer

$$-7$$

Question 4

Consider $$f(x) = x^2$ - 6x + 7 on$$ $$[0, 10]. $$What is the absolute minimum value of $$f(x) on $$this interval?

Accepted Answer

$$-2$$

Question 5

For $$f(x) = x^2$ - 6x + 7 on$$ $$[0, 10]$$, what is the absolute maximum value?

Accepted Answer

$$47$$

Question 6

Given $$f(x) = x^2$ - 6x + 7 on$$ $$[4, 10]$$, what is the absolute minimum value?

Accepted Answer

$$3$$

Question 7

For $$f(x) = x^2$ - 6x + 7 on$$ $$[4, 10]$$, what is the absolute maximum value?

Accepted Answer

$$47$$

Question 8

A business models its profit with $$f(x) = x^3$ - 6x^2$ + 9x - 6$$ for $$x in$$ $$[-1, 5]. $$What is the absolute minimum value of $$f(x) on $$this interval?

Accepted Answer

$$-12$$

Question 9

For $$f(x) = x^3$ - 6x^2$ + 9x - 6 on$$ $$[-1, 5]$$, what is the absolute maximum value?

Accepted Answer

$$8$$

Question 10

A company uses $$f(x) = 2x^3$ - 3x^2$ - 12x + 1 to $$model production costs for $$x in$$ $$[-3, 1]. $$What is the absolute maximum value of $$f(x) on $$this interval?

Accepted Answer

$$22$$

Question 11

For $$f(x) = 2x^3$ - 3x^2$ - 12x + 1 on$$ $$[-3, 1]$$, what is the absolute minimum value?

Accepted Answer

$$-20$$

Question 12

Which of the following statements is TRUE regarding absolute extrema on a closed interval?

Accepted Answer

Absolute extrema can occur at endpoints or critical points within the interval.

Question 13

If $$f(x) is $$continuous on $$[a, b]$$, which theorem guarantees that $$f(x)$$ attains both an absolute maximum and minimum on $$[a, b]$$?

Accepted Answer

Extreme Value Theorem

Question 14

A company wants to maximize its revenue modeled by $$f(x) = 9 - x^2$ for x$ in $$[-4, 4]$$. At what value of x$ does the absolute maximum occur?

Accepted Answer

$$x = 0$$

Question 15

For $$f(x) = x^2$ - 6x + 7$$, what is the $$x$$ value where the absolute minimum occurs on $$[0, 10]$$?

Accepted Answer

$$x = 3$$

Question 16

A business models its profit with $$f(x) = x^3$ - 6x^2$ + 9x - 6$$ for $$x in$$ $$[-1, 5]. At $$what value of $$x$$ does the absolute minimum occur?

Accepted Answer

$$x = -1$$

Question 17

For $$f(x) = 2x^3$ - 3x^2$ - 12x + 1 on$$ $$[-3, 1]$$, at which $$x$$ value does the absolute maximum occur?

Accepted Answer

$$x = -3$$

Question 18

If a function $$f(x)$$ has its absolute minimum at $$x = a$$ and absolute maximum at $$x = b on$$ $$[a, b]$$, which of the following is necessarily true?

Accepted Answer

$$f(a) is $$the smallest value and $$f(b) is $$the largest value of $$f(x) on$$ $$[a, b]$$

Question 19

A company wants to minimize its production cost modeled by $$f(x) = 2x^3$ - 3x^2$ - 12x + 1$$ for $$x in$$ $$[-3, 1]. At $$what value of $$x$$ does the absolute minimum occur?

Accepted Answer

$$x = 1$$

Business Calculus: Finding Absolute Maximum and Minimum Values on Closed Intervals

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