Q: If $$f(x)$$ has a local maximum at $$x = a$$, which of the following must be true?

$$f'(a) = 0$$ \,\text{or} \, f'(a) \,\text{is undefined}$

Question 1

Given the function $$f(x) = \frac{1}{x}$$, which of the following is a partition number for $$f'(x)$$?

Accepted Answer

$$x = 0$$

Question 2

For the function $$f(x) = \frac{1}{x}$$, which of the following statements is true about its critical numbers?

Accepted Answer

There are no critical numbers because $$f'(x)$$ does not equal zero anywhere in its domain.

Question 3

A company’s total cost function for producing $$x$$ camp stoves per week is $$C(x). If $$the graph of $$C'(x) is $$positive and increasing, what does this indicate about the efficiency of the production process?

Accepted Answer

Marginal costs are increasing, indicating decreasing efficiency as more stoves are produced.

Question 4

Suppose the graph of $$y = f(x)$$ has local maxima at $$x = 2$$ and $$x = 5. $$Which of the following lists all $$x-$$coordinates where $$f(x)$$ has a local maximum?

Accepted Answer

$$x = 2, 5$$

Question 5

If $$f(x) is $$continuous everywhere except at $$x = 0$$, and $$f'(x) is $$undefined at $$x = 0$$, what is $$x = 0$$ called in the context of partition numbers?

Accepted Answer

A partition number for $$f'(x)$$

Question 6

A manufacturer observes that the marginal cost function $$C'(x) is $$positive and decreasing as production increases. What does this suggest about the production process?

Accepted Answer

Marginal costs are decreasing, indicating increasing efficiency.

Question 7

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

Accepted Answer

A value $$x$$ where $$f'(x) = 0 or$$ $$f'(x) is $$undefined, and $$x is in $$the domain of $$f(x).$$

Question 8

If $$f(x)$$ has a local maximum at $$x = a$$, which of the following must be true?

Accepted Answer

$$f'(a) = 0$$ \,	ext{or} \, f'(a) \,	ext{is undefined}$

Question 9

A company’s cost function is $$C(x) = 100,000 + 50x. $$What is the marginal cost, and what does it represent?

Accepted Answer

$$C'(x) = 50$$, representing the cost to produce one additional camp stove.

Question 10

Suppose $$f(x) is $$differentiable everywhere except at $$x = 0$$, where $$f(x) is $$undefined. Which of the following is true about $$x = 0$$?

Accepted Answer

$$x = 0 is a $$partition number for $$f'(x)$$ but not a critical number for $$f(x).$$

Question 11

If the marginal cost function $$C'(x) is $$negative, what does this imply about the cost of producing additional units?

Accepted Answer

Producing additional units reduces the total cost.

Question 12

A company’s cost function $$C(x) is $$shown to be convex upward. What does this indicate about the marginal cost $$C'(x)$$?

Accepted Answer

$$C'(x) is $$increasing as $$x$$ increases.

Question 13

If $$f(x)$$ has a local maximum at $$x = 3$$, which of the following must be true about $$f'(x) at$$ $$x = 3$$?

Accepted Answer

$$f'(3) = 0$$ \,	ext{or} \, f'(3) \,	ext{is undefined}$

Question 14

Which of the following is NOT a possible reason for a value $$x = a to be a $$partition number for $$f'(x)$$?

Accepted Answer

$$f(x) is $$differentiable and $$f'(x) is $$continuous at $$x = a.$$

Question 15

A company’s marginal cost function $$C'(x) is $$positive and decreasing. What does this mean for the company’s production efficiency?

Accepted Answer

Efficiency is increasing as marginal costs decrease.

Question 16

If $$f(x) is $$continuous and differentiable everywhere except at $$x = 0$$, where $$f(x) is $$undefined, which of the following is true?

Accepted Answer

$$x = 0 is a $$partition number for $$f'(x).$$

Question 17

Suppose $$C(x) is $$the total cost function for a company, and $$C'(x) is $$always positive. What does this imply about the cost of producing additional units?

Accepted Answer

Producing additional units always increases the total cost.

Question 18

If $$f(x)$$ has a critical number at $$x = a$$, which of the following must be true?

Accepted Answer

$$f'(a) = 0$$ \,	ext{or} \, f'(a) \,	ext{is undefined}$

Business Calculus: Derivatives, Critical Points, and Marginal Cost Analysis

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