Question 1

Given the graph in question 1, at which $$x-$$value does the function attain its absolute minimum?

Accepted Answer

$$x = 0$$

Question 2

For the function $$f(x) = -3x^4$ + 2x^3$ - 36x^2$ - 8$$, which of the following is a critical point?

Accepted Answer

$$x = -3$$

Question 3

Given $$f(x) = \frac{x^2$}{3x+6}$$, for $$x \u003e -2$$, which of the following statements is true about its extrema?

Accepted Answer

It has a relative minimum at $$x = 0$$

Question 4

For $$f(x) = 5x - 4x$$, $$x \u003e 0$$, what is the value of $$x$$ where the function attains its absolute maximum?

Accepted Answer

There is no absolute maximum

Question 5

If $$f(x) = 6x \ln x$$, what is the $$x-$$value of its critical point?

Accepted Answer

$$x = 1$$

Question 6

Given the profit function $$P(x) = -0.3x^3$ + \frac{27}{2}x^2$ - 60x + 100$$, where $$x is $$the number of units sold (in thousands), what is the production level $$x$$ that maximizes profit?

Accepted Answer

$$x = 15$$

Question 7

Which of the following best describes an absolute maximum of a function $$f(x)$$?

Accepted Answer

A point where $$f(x) is $$greater than or equal to all points in the domain

Question 8

Suppose a company’s cost function is $$C(x) = 2x^2$ - 12x + 30. $$What is the production level $$x$$ that minimizes cost?

Accepted Answer

$$x = 3$$

Question 9

If $$f(x)$$ has a critical point at $$x = a$$ and $$f''(a) \u003e 0$$, what can you conclude about $$f(x) at$$ $$x = a$$?

Accepted Answer

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

Question 10

A pharmaceutical company models the revenue from a new drug as $$R(x) = 100x - 2x^2$, where x$ is the number of units sold (in thousands). What is the maximum possible revenue?

Accepted Answer

$$R = 2500$$

Question 11

Which of the following is NOT a necessary condition for a point $$x = c to be a $$local extremum of $$f(x)$$?

Accepted Answer

$$f''(c) 
eq 0$$

Question 12

A company’s profit function is $$P(x) = -2x^2$ + 40x - 100. $$What is the maximum profit, and at what value of $$x$$ does it occur?

Accepted Answer

Maximum profit is $$-100 at$$ $$x = 10$$

Question 13

If $$f(x) = x^3$ - 6x^2$ + 9x + 1$$, what is the $$x-$$coordinate of the relative minimum?

Accepted Answer

$$x = 2$$

Question 14

A business wants to maximize the area of a rectangular enclosure with 100 meters of fencing. What is the maximum area?

Accepted Answer

$$2500$$ square meters

Question 15

Given $$f(x) = x^4$ - 4x^2$, which of the following is a relative maximum?

Accepted Answer

$$x = 0$$ and $$f(0) = 0$$

Question 16

If $$f(x) is $$continuous on $$[a, b]$$ and differentiable on $$(a, b)$$, which theorem guarantees that $$f(x)$$ attains both an absolute maximum and minimum on $$[a, b]$$?

Accepted Answer

Extreme Value Theorem

Question 17

A company’s revenue function is $$R(x) = 50x - x^2$, and its cost function is C(x) = 10x + 20$. What is the production level x$ that maximizes profit?

Accepted Answer

$$x = 10$$

Question 18

If $$f(x) = x^2$ - 4x + 3$$, what is the minimum value of $$f(x)$$?

Accepted Answer

$$f(2) = -1$$

Question 19

A company’s marginal cost is $$C'(x) = 6x - 12. At $$what production level is the cost minimized?

Accepted Answer

$$x = 2$$

Question 20

If $$f(x)$$ has a local maximum at $$x = 1$$ and a local minimum at $$x = 3$$, which of the following must be true?

Accepted Answer

$$f'(1) = 0$$ and $$f'(3) = 0$$

Extrema and Optimization in Business Calculus

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