Question 1

Suppose the profit function for a company is given by $$P(x) = R(x) - C(x)$$, where $$R(x) is $$the revenue and $$C(x) is $$the cost for producing $$x$$ units. If $$P'(x) = 0$$ and $$P''(x) \u003c 0 at$$ $$x = a$$, what does this indicate about the profit at $$x = a$$?

Accepted Answer

Profit is maximized at $$x = a.$$

Question 2

A pharmaceutical company models its revenue with $$R(x) = 100x - x^2$, where x$ is the number of units sold. What is the number of units that maximizes revenue?

Accepted Answer

$$x = 50$$

Question 3

If $$f'(x) \u003e 0$$ and $$f''(x) \u003c 0 on an $$interval, what can you conclude about the function $$f(x) on $$that interval?

Accepted Answer

$$f(x) is $$increasing and concave down.

Question 4

A company’s cost function is $$C(x) = 500 + 20x. $$What is the marginal cost at $$x = 10$$?

Accepted Answer

$$20$$

Question 5

Given $$f(x)$$ has a critical point at $$x = 3$$ and $$f''(3) \u003e 0$$, what does this tell you about $$f(x) at$$ $$x = 3$$?

Accepted Answer

$$f(x)$$ has a local minimum at $$x = 3.$$

Question 6

If $$f'(x) = 0$$ and $$f''(x) = 0 at$$ $$x = a$$, which of the following is true?

Accepted Answer

The second derivative test is inconclusive at $$x = a.$$

Question 7

A manufacturer finds that the marginal revenue $$R'(x) is $$less than the marginal cost $$C'(x)$$ for all $$x \u003e 100. $$What should the manufacturer do to maximize profit?

Accepted Answer

Decrease production below $$x = 100.$$

Question 8

If $$f''(x)$$ changes sign at $$x = c$$, what does this indicate about $$f(x) at$$ $$x = c$$?

Accepted Answer

$$x = c is an $$inflection point.

Question 9

A business models its profit as $$P(x) = -2x^2$ + 40x - 100. $$What is the maximum profit and at what value of $$x$$ does it occur?

Accepted Answer

Maximum profit is $$-2(10)^2 + 40(10) - 100 = 100$ at x = 10$.

Question 10

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 11

A company’s revenue function is $$R(x) = 200x - 5x^2$. What is the marginal revenue when x = 8$?

Accepted Answer

$$R'(x) = 200 - 10x$$, so $$R'(8) = 200 - 10(8) = 120$$

Question 12

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

Accepted Answer

$$f'(x)  0$$

Question 13

A hospital models its cost for patient care as $$C(x) = 1000 + 50x + 0.5x^2$. What is the marginal cost when x = 20$?

Accepted Answer

$$C'(x) = 50 + x$$, so $$C'(20) = 70$$

Question 14

If $$f'(x) is $$increasing and $$f''(x) \u003e 0$$, what does this indicate about the graph of $$f(x)$$?

Accepted Answer

The graph is concave up and $$f(x) is $$increasing at an increasing rate.

Question 15

A company’s profit function is $$P(x) = -x^2$ + 12x - 20. $$Find the value of $$x$$ that maximizes profit and the maximum profit.

Accepted Answer

$$x = 6$$, $$P(6) = -36 + 72 - 20 = 16$$

Question 16

If $$f(x)$$ has a local maximum at $$x = 2$$, which of the following must be true?

Accepted Answer

$$f'(2) = 0$$ and $$f''(2) \u003c 0$$

Question 17

A business finds that $$R'(x) = C'(x) at$$ $$x = 15. $$What does this mean in terms of profit maximization?

Accepted Answer

Profit is maximized at $$x = 15.$$

Question 18

If $$f'(x) is $$negative and $$f''(x) is $$negative on an interval, what does this indicate about $$f(x)$$?

Accepted Answer

$$f(x) is $$decreasing and concave down.

Question 19

A company’s cost function is $$C(x) = 300 + 15x + x^2$. What is the marginal cost at x = 5$?

Accepted Answer

$$C'(x) = 15 + 2x$$, so $$C'(5) = 25$$

Applications of Derivatives: First and Second Derivative Tests, Optimization, and Revenue/Profit Problems

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