Question 1

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

Accepted Answer

$$f(0) = 9$$

Question 2

For the function $$f(x) = 9 - x^2$ on $$[-4, 4]$$, at which x$-value does the absolute minimum occur?

Accepted Answer

$$x = 4$$

Question 3

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

Accepted Answer

$$f(3) = -2$$

Question 4

For $$f(x) = x^2$ - 6x + 7 on$$ $$[4, 10]$$, where does the absolute maximum occur?

Accepted Answer

$$x = 10$$

Question 5

Given $$f(x) = x^3$ - 6x^2$ + 9x - 6 on$$ $$[-1, 5]$$, which of the following is the absolute minimum value?

Accepted Answer

$$f(-1) = -22$$

Question 6

For the function $$f(x) = x^3$ - 6x^2$ + 9x - 6 on$$ $$[-1, 5]$$, at which $$x-$$value does the absolute maximum occur?

Accepted Answer

$$x = 3$$

Question 7

A business models its profit with $$f(x) = 2x^3$ - 3x^2$ - 12x + 1$$ for $$x in$$ $$[-3, 1]$$, where $$x is $$the number of units produced (in thousands). What is the absolute maximum profit on this interval?

Accepted Answer

$$f(0) = 1$$

Question 8

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

Accepted Answer

$$x = -3$$

Question 9

Which of the following statements is TRUE about absolute extrema for a continuous function $$f(x) on a $$closed interval $$[a, b]$$?

Accepted Answer

Absolute extrema occur at critical points or endpoints.

Question 10

A company’s cost function is $$C(x) = x^2$ - 6x + 7$$ for $$x in$$ $$[0, 10]. $$What is the minimum cost, and at what production level does it occur?

Accepted Answer

Minimum cost is $$-2 at$$ $$x = 3$$

Question 11

Suppose $$f(x) = 9 - x^2$ models the height (in meters) of a projectile at time x$ seconds, for x$ in $$[-4, 4]$$. What is the maximum height reached by the projectile?

Accepted Answer

$$9$$ meters$

Question 12

If $$f(x) = x^3$ - 6x^2$ + 9x - 6$$ models the profit (in thousands of dollars) for producing $$x$$ units, what is the maximum profit on $$[-1, 5]$$?

Accepted Answer

$$6$$ thousand dollars

Question 13

Which of the following is the correct process to find the absolute extrema of a continuous function $$f(x) on$$ $$[a, b]$$?

Accepted Answer

Find critical points, evaluate $$f at $$critical points and endpoints, then compare values.

Question 14

A manufacturer’s revenue is modeled by $$f(x) = 2x^3$ - 3x^2$ - 12x + 1$$ for $$x in$$ $$[-3, 1]. $$What is the lowest revenue on this interval?

Accepted Answer

$$-80$$

Question 15

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

Accepted Answer

$$f(4) = 3$$

Question 16

A company wants to maximize its profit, modeled by $$f(x) = x^3$ - 6x^2$ + 9x - 6 on$$ $$[-1, 5]. $$Which $$x-$$value should they choose?

Accepted Answer

$$x = 3$$

Question 17

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

Accepted Answer

$$f(c) \leq f(x)$$ for all $$x in$$ $$[a, b]$$

Absolute Extrema on Closed Intervals – Business Calculus Guidance

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