Question 1

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

Accepted Answer

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

Question 2

Suppose $$f(x)$$ has a relative maximum at $$x = a. $$Which of the following must be true at $$x = a$$?

Accepted Answer

$$f'(a) = 0$$ and $$f''(a) \u003c 0$$

Question 3

A function $$f(x) is $$concave down on an interval if:

Accepted Answer

$$f''(x) \u003c 0$$ for all $$x in $$the interval.

Question 4

Which of the following is the correct constraint equation for a rectangular garden with perimeter 300 meters?

Accepted Answer

$$2x + 2y = 300$$

Question 5

Given the area function $$A(x) = 150x - x^2$ for a rectangular garden, what value of x$ maximizes the area?

Accepted Answer

$$x = 75$$

Question 6

If $$f'(a) = 0$$ and $$f''(a) \u003e 0$$, what does this indicate about $$f(x) at$$ $$x = a$$?

Accepted Answer

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

Question 7

A swimming club offers memberships at $$p = 200$$ for at least 100 people, and for each additional member, the price drops by $$1. $$What is the demand equation relating price $$p$$ and number of memberships $$x$$?

Accepted Answer

$$p = -x + 300$$

Question 8

For the revenue function $$R(x) = -x^2$ + 300x$$, what value of $$x$$ maximizes revenue?

Accepted Answer

$$x = 150$$

Question 9

A monopolist faces the demand equation $$p = 200 - 3x$$ and cost function $$C(x) = 75 + 80x - x^2$. What is the profit-maximizing quantity x$?

Accepted Answer

$$x = 30$$

Question 10

If the government imposes a tax of $$4$$ per unit on the monopolist, what is the new profit-maximizing price per unit?

Accepted Answer

$$p = 113$$

Question 11

Which of the following is true about inflection points of a function $$f(x)$$?

Accepted Answer

They occur where $$f''(x)$$ changes sign and $$f(x) is $$continuous.

Question 12

For the cost minimization problem with $$C = 1500/y + 15y$$ and area constraint $$xy = 75$$, what value of $$y$$ minimizes cost?

Accepted Answer

$$y = 10$$

Question 13

Which of the following best describes a vertical asymptote of a function?

Accepted Answer

A vertical line $$x = a$$ where the denominator of the function equals zero and the function is undefined.

Question 14

Given the surface area function $$A(x) = 2x^2$ + 500/x$$ for a wind shelter with volume constraint $$x^2y = 250$, what value of x$ minimizes the surface area?

Accepted Answer

$$x = 5$$

Question 15

If $$f'(x)$$ changes from positive to negative at $$x = a$$, what does this indicate about $$f(x) at$$ $$x = a$$?

Accepted Answer

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

Question 16

Which of the following is the correct objective equation for maximizing the area of a rectangle with perimeter $$P$$?

Accepted Answer

$$A = xy$$

Question 17

For the function $$f(x) = x^3$ - x + 2$$, what is the value of $$x$$ where the inflection point occurs?

Accepted Answer

$$x = 0$$

Question 18

Which of the following is NOT a step in solving an optimization problem?

Accepted Answer

Set the second derivative equal to zero and solve.

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

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