Question 1

Given the function $$f(x) = 15x^4$ - 4x^3$ on the interval $$[-2, 2]$$, what is the absolute maximum value and at which x$-value does it occur?

Accepted Answer

$$f(-2) = 272 at$$ $$x = -2$$

Question 2

For the function $$h(x) = x e$$^{-5x}$$, what is the x$-coordinate of the relative extremum?

Accepted Answer

$$x = 0$$

Question 3

A company’s revenue and cost functions are $$R(x) = 3x$$ and $$C(x) = 0.05x^2$ + 0.4x + 1. $$What is the production level $$x$$ that maximizes profit?

Accepted Answer

$$x = 26$$

Question 4

Given $$g(x) = x^3$ - x^2$ + 4x$$, find the coordinates of the inflection point.

Accepted Answer

$$(\frac{1}{2}, g(\frac{1}{2}))$$

Question 5

For the function $$f(x) = \ln(x^2$ + 2x + 4)$$, what is the critical value?

Accepted Answer

$$x = -1$$

Question 6

A car’s fuel economy $$E(s) in $$mpg is maximized at a certain speed $$s. If$$ $$E(s) = -0.05s^2$ + 4s + 10$$, at what speed is the maximum fuel economy obtained?

Accepted Answer

$$s = 40$$ mph

Question 7

Which of the following best describes a point of inflection for a function $$f(x)$$?

Accepted Answer

A point where $$f''(x)$$ changes sign

Question 8

Given the function $$h(x) = x^3$ + \frac{17}{2}x^2$ + 10x + 1$$, what are the $$x-$$values where relative extrema may occur?

Accepted Answer

$$x = -4, x = 0$$

Question 9

If the graph of $$f'(x) is $$negative on $$(3,8)$$ and positive on $$(8,\infty)$$, what can you say about $$f(x)$$?

Accepted Answer

$$f(x)$$ has a relative minimum at $$x = 8$$

Question 10

A productivity function is $$M(t) = -3t^2$ + 150t + 120$$ for $$0 \leq t \leq 40. In $$which year is the maximum productivity achieved?

Accepted Answer

$$t = 25$$

Question 11

For the function $$f(x) = x^4$ - 4x^3$ + 6x^2$ - 4x + 1$$, what is the interval where the function is concave up?

Accepted Answer

$$(-\infty, 1)$$ and $$(3, \infty)$$

Question 12

Suppose the cost to produce $$x$$ chairs is $$C(x) = 0.006x^3$ + 0.07x^2$ + 15x + 600. $$What is the marginal cost when $$x = 40$$?

Accepted Answer

$$C'(40) = 0.018(40)^2$ + 0.14(40) + 15$$

Question 13

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

Accepted Answer

A point where $$f(x) is $$not defined

Question 14

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

Accepted Answer

The graph is concave up

Question 15

Given $$f(x) = x^3$ - 3x^2$ + 2x$$, which of the following is true about its relative extrema?

Accepted Answer

There is a relative maximum at $$x = 1$$ and a relative minimum at $$x = 2$$

Question 16

For the function $$f(x) = x^4$ - 4x^3$ + 6x^2$ - 4x + 1$$, at which $$x-$$value does the inflection point occur?

Accepted Answer

$$x = 1$$

Calculus Chapter 3 Review: Critical Points, Extrema, Concavity, and Applications

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