Question 1

Let $$f(x) = 15x^4$ - 4x^3$ on the interval $$[-2, 2]$$. What is the absolute maximum value of f(x$$)$$ on this interval, and at which x$-value does it occur?

Accepted Answer

$$f(-2) = 208 at$$ $$x = -2$$

Question 2

Suppose the cost to produce $$x$$ chairs per month is $$C(x) = 0.006x^3$ + 0.07x^2$ + 15x + 600. $$What is the marginal cost of producing the 40th chair?

Accepted Answer

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

Question 3

Given $$h(x) = xe$$^{5x}$$, find all critical values.

Accepted Answer

There are no critical values.

Question 4

For the function $$M(t) = -3t^2$ + 150t + 120$$, $$0 \leq t \leq 40$$, representing monthly productivity, in which year is the maximum productivity achieved?

Accepted Answer

$$t = 25$$

Question 5

A function $$f is $$concave down on $$(1,4)$$ and concave up on $$(5,6). $$Which of the following intervals could contain a point of inflection?

Accepted Answer

$$(4,5)$$

Question 6

Let $$g(x) = \ln(x^2$ + 2x + 4). $$Find any critical values.

Accepted Answer

Critical value at $$x = -1$$

Question 7

Given $$R(x) = 3x$$ and $$C(x) = 0.05x^2$ + 0.4x + 1$$, what is the production level for maximum profit?

Accepted Answer

$$x = 20$$

Question 8

For $$f(x) = x^3$ + \frac{17}{2}x^2$ + 10x + 1$$, what are the $$x-$$values of the relative extrema?

Accepted Answer

$$x = -4$$ and $$x = 0$$

Question 9

Which of the following best describes the critical value of a function?

Accepted Answer

A value where the derivative is zero or does not exist.

Question 10

Suppose the graph of $$f'(x) is $$negative on $$(3,8)$$ and positive on $$(5,\infty). $$Where is $$f(x)$$ increasing?

Accepted Answer

$$(5,\infty)$$

Question 11

For $$h(x) = xe$$^{5x}$$, on which interval is h(x$$)$$ increasing?

Accepted Answer

$$\left(-\frac{1}{5}, \infty\right)$$

Question 12

A car's fuel economy in mpg is modeled by $$M(s) = -0.85s^2$ + 42s + 15$$, where $$s is $$speed in mph. At what speed is the absolute maximum fuel economy obtained?

Accepted Answer

$$s = 25$$ mph

Question 13

Given $$f(x) = x^3$ + \frac{17}{2}x^2$ + 10x + 1$$, on which intervals is $$f(x)$$ increasing?

Accepted Answer

$$(-\infty, -4)$$ and $$(0, \infty)$$

Question 14

For $$g(x) = x^4$ - x^3$ + 4x^2$ + 6$$, what are the coordinates of the relative extrema?

Accepted Answer

Relative minimum at $$x = 0$$, relative maximum at $$x = 3$$

Question 15

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

Accepted Answer

$$f(x) is $$concave up

Question 16

For $$f(x) = x^3$ + \frac{17}{2}x^2$ + 10x + 1$$, what are the $$x-$$values of the inflection points?

Accepted Answer

$$x = -\frac{17}{6}$$

Question 17

Suppose $$f'(x)$$ changes from positive to negative at $$x = a. $$What does this indicate about $$f(x) at$$ $$x = a$$?

Accepted Answer

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

Question 18

Given $$C(x) = 0.006x^3$ + 0.07x^2$ + 15x + 600$$, what is the actual cost of producing 42 chairs per month?

Accepted Answer

$$C(42) = 0.006(42)^3$ + 0.07(42)^2$ + 15(42) + 600$$

Question 19

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

Accepted Answer

$$x = c is a $$point of inflection

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

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