Q: A function $$f(x)$$ has a relative maximum at $$x = a if$$:

$$f'(a) = 0$$ \,\text{and}\, f''(a) \u003c 0$

Q: A point $$x = a is an $$inflection point of $$f(x) if$$:

$$f''(a) = 0$$ \text{ and the sign of } f''(x) \text{ changes at } x = a$

Q: A company wants to maximize the area of a rectangular garden with 300 meters of fencing. What are the dimensions of the garden with the largest area?

$$75 \text{ m} \times 75 \text{ m}$$

Question 1

Which of the following statements best describes when a function $$f(x) is $$increasing on an interval?

Accepted Answer

When $$f'(x) \u003e 0$$ for all $$x in $$the interval.

Question 2

A function $$f(x)$$ has a relative maximum at $$x = a if$$:

Accepted Answer

$$f'(a) = 0$$ \,	ext{and}\, f''(a) \u003c 0$

Question 3

Which of the following is NOT a necessary step in the general approach to curve sketching?

Accepted Answer

Find the area under the curve.

Question 4

Suppose $$f'(x)$$ changes from negative to positive at $$x = a. $$What does this indicate about $$f(x) at$$ $$x = a$$?

Accepted Answer

$$f(x)$$ has a relative minimum at $$x = a$$

Question 5

A point $$x = a is an $$inflection point of $$f(x) if$$:

Accepted Answer

$$f''(a) = 0$$ 	ext{ and the sign of } f''(x) 	ext{ changes at } x = a$

Question 6

Which of the following best describes a horizontal asymptote of a function $$f(x)$$?

Accepted Answer

A straight, horizontal line that the graph approaches as $$x$$ increases or decreases without bound.

Question 7

Given $$f(x) = x^3$ - 3x^2$ + 2$$, find all critical points.

Accepted Answer

$$x = 1$$ and $$x = 2$$

Question 8

A company wants to maximize the area of a rectangular garden with 300 meters of fencing. What are the dimensions of the garden with the largest area?

Accepted Answer

$$75 	ext{ m} 	imes 75 	ext{ m}$$

Question 9

If $$f''(x) \u003e 0$$ for all $$x in an $$interval, what can you conclude about the graph of $$f(x) on $$that interval?

Accepted Answer

The graph is concave up.

Question 10

A profit function is given by $$P(x) = -x^2$ + 6x + 15. $$What is the value of $$x$$ that maximizes profit?

Accepted Answer

$$x = 3$$

Question 11

A bookstore sells 8000 books per year. The ordering cost per order is $$40$$, and the carrying cost per book is $$2. $$What is the economic order quantity (EOQ) that minimizes total inventory cost?

Accepted Answer

$$x = 400$$

Question 12

If $$f'(x) = 0$$ and $$f''(x) \u003c 0 at$$ $$x = a$$, what does this tell you about $$f(x) at$$ $$x = a$$?

Accepted Answer

$$f(x)$$ has a relative maximum at $$x = a$$

Question 13

A swimming club offers memberships at $$p = 200 - x$$, where $$x is $$the number of memberships sold above 100. What number of memberships maximizes revenue?

Accepted Answer

$$x = 150$$

Question 14

Which of the following is true about the relationship between the first and second derivatives and the shape of the graph?

Accepted Answer

If $$f'(x)  0$$, $$f(x) is $$decreasing and concave up.

Question 15

A function $$f(x)$$ has a critical point at $$x = 2. $$The first derivative changes from positive to negative at $$x = 2. $$What does this mean?

Accepted Answer

$$f(x)$$ has a relative maximum at $$x = 2$$

Chapter 2: Applications of the Derivative – Business Calculus Study Notes

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