Question 1

Given the graph of a function $$f(x)$$ defined on the closed interval $$[0,4] ($$see the first image), what are the absolute maximum and minimum values of $$f(x) on $$this interval?

Accepted Answer

Absolute maximum at $$x=4$$, $$f(4) = 5$$; absolute minimum at $$x=2$$, $$f(2) = -5$$

Question 2

Which of the following statements best describes the Extreme-Value Theorem?

Accepted Answer

A continuous function defined on a closed interval $$[a, b]$$ must have an absolute maximum and minimum on $$[a, b].$$

Question 3

Given $$f(x) = x^2$ - 12x on$$ $$[-1, 5]$$, what is the absolute maximum value?

Accepted Answer

$$f(-1) = 13$$

Question 4

For the function $$f(x) = x^3$ - 4 on$$ $$[-1, 2]$$, where does the absolute minimum occur?

Accepted Answer

At $$x = -1$$

Question 5

Suppose $$f(x) = -x^2$ + 6x - 3 on$$ $$(-\infty, \infty). $$According to the Critical Point Theorem, what is the absolute maximum value?

Accepted Answer

$$f(3) = 6$$

Question 6

A population model for fish is $$f(S) = \frac{100S}{4+S}$$, where $$S is $$the number of adults in thousands. What is the formula for the harvest $$H$$?

Accepted Answer

$$H = f(S) - S$$

Question 7

If $$f(S) = 45S$$^{2/3}$$, what value of S$ maximizes the sustainable harvest H(S) = f(S) - S$?

Accepted Answer

$$S = 27$$

Question 8

A farmer has 2400 meters of fencing to build a rectangular pen with one fence across its width. What is the maximum area that can be enclosed?

Accepted Answer

$$200,000 m$^2$$

Question 9

Given $$f(x) = \frac{x^2$ + 36}{2x}$$ for $$1 \leq x \leq 12$$, when is the selenium in the soil minimized?

Accepted Answer

At $$x = 6$$

Question 10

Which of the following is NOT a possible location for an absolute extremum of a continuous function on $$[a, b]$$?

Accepted Answer

At a point where the function is not defined

Question 11

If $$f(x) is $$continuous on $$[a, b]$$ and has only one critical value $$c in$$ $$(a, b)$$, and $$f''(c) \u003e 0$$, what can you conclude?

Accepted Answer

$$f(c) is an $$absolute minimum

Question 12

A company wants to maximize profit, which is modeled by $$P(x) = -2x^2$ + 8x + 10$$ for $$0 \leq x \leq 5. $$What is the maximum profit?

Accepted Answer

$$P(2) = 18$$

Question 13

Which of the following best describes a critical value of a function $$f(x)$$?

Accepted Answer

A value where $$f'(x) = 0 or$$ $$f'(x)$$ does not exist

Question 14

If $$f(x) is $$continuous on $$[a, b]$$ and differentiable on $$(a, b)$$, which steps are necessary to find the absolute extrema?

Accepted Answer

Find $$f'(x)$$, solve $$f'(x) = 0$$, evaluate $$f(x) at $$critical points and endpoints

Absolute Extrema and Applications of Extrema (Business Calculus Study Notes)

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