Q: A function $$f(x) is $$increasing on an interval if:

$$f'(x) \u003e 0$$ \text{ for all } x \text{ in the interval}$

Question 1

Suppose a company’s profit function is given by $$P(x) = -2x^2$ + 40x - 100$$, where $$x is $$the number of units sold. Using the First-Derivative Test, at what value of $$x$$ does the company achieve maximum profit?

Accepted Answer

$$x = 10$$

Question 2

Given the function $$f(x) = x^3$ - 6x^2$ + 9x + 1$$, use the Second-Derivative Test to classify the critical point at $$x = 1.$$

Accepted Answer

Relative maximum

Question 3

A hotel owner rents $$y$$ rooms per night when the price is $$x$$ dollars per room, with $$y = 200 - 2x. $$What price per room maximizes the revenue?

Accepted Answer

$$x = 50$$

Question 4

For the function $$f(x) = x^4$ - 4x^3$, find all critical values.

Accepted Answer

$$x = 0$$ 	ext{ and } x = 3$

Question 5

Which of the following statements is TRUE about the concavity of a function $$f(x)$$?

Accepted Answer

If $$f''(x) \u003e 0$$, the graph is concave up at $$x.$$

Question 6

Given the demand function $$q = D(x) = 1200 - 3x$$, what is the elasticity of demand $$E(x)$$?

Accepted Answer

$$E(x) = \dfrac{-3x}{1200 - 3x}$$

Question 7

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

Accepted Answer

$$f'(x) \u003e 0$$ 	ext{ for all } x 	ext{ in the interval}$

Question 8

Which of the following is NOT a critical value of $$f(x)$$?

Accepted Answer

A point where $$f'(x) is $$positive

Question 9

Suppose $$f(x)$$ has a critical value at $$x = c$$ and $$f''(c) = 0. $$What should you do to classify the extremum?

Accepted Answer

Use the First-Derivative Test

Question 10

A company’s cost function is $$C(x) = 0.035x^2$ + 500x + 12,500. $$What is the average cost per case when 100 cases are produced?

Accepted Answer

$$C_{avg} = \dfrac{0.035(100)^2$ + 500(100) + 12,500}{100}$$

Question 11

If $$E(x) = 1$$ for a demand function, what does this mean about total revenue?

Accepted Answer

Total revenue is maximized

Question 12

Given $$f(x) = x^3$ - 3x^2$ + 2$$, find the intervals where $$f(x) is $$increasing.

Accepted Answer

$$(2, \infty)$$ and $$(0, 2)$$

Question 13

Which of the following best describes a point of inflection?

Accepted Answer

A point where the concavity of the function changes

Question 14

For the function $$f(x) = x^2$ + 4x + 5 on $$the interval $$[-2, 1]$$, what is the absolute minimum value?

Accepted Answer

$$f(-2) = 5$$

Question 15

A retailer finds that the demand for a product is given by $$q = 200e$$^{-0.01x}$$, where x$ is the price. What is the elasticity of demand E(x$$)$$?

Accepted Answer

$$E(x) = -0.01x$$

Question 16

If $$f'(x) = 0$$ and $$f''(x) \u003c 0 at$$ $$x = c$$, what can you conclude about $$f(x) at$$ $$x = c$$?

Accepted Answer

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

Question 17

Which of the following is the correct process for finding absolute extrema on a closed interval $$[a, b]$$?

Accepted Answer

Find critical values, evaluate $$f(x) at $$critical values and endpoints, compare values

Question 18

A company’s revenue function is $$R(x) = 50x - 0.5x^2$ and cost function is C(x) = 10x + 3$. What is the profit-maximizing number of units?

Accepted Answer

$$x = 40$$

Question 19

If $$f(x) is $$decreasing and concave up on an interval, which of the following is true?

Accepted Answer

$$f'(x)  0$$

Applications of Differentiation: Extrema, Concavity, Optimization, and Elasticity

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