Question 1

Which of the following is the correct definition of a critical number for a function $$f$$?

Accepted Answer

A value $$c in $$the domain of $$f$$ where $$f'(c) = 0 or$$ $$f'(c)$$ does not exist.

Question 2

Given the function $$f(x) = x^3$ - 2x^2$ + 7$$, find all critical numbers.

Accepted Answer

$$x = 1$$

Question 3

For the function $$f(x) = 2x^2$ - 8x + 12$$, what are the critical numbers?

Accepted Answer

$$x = 2$$

Question 4

Suppose $$f'(x) \u003e 0$$ for all $$x in an $$interval. What can you conclude about $$f(x) on $$that interval?

Accepted Answer

$$f(x) is $$increasing.

Question 5

Suppose $$f'(x) \u003c 0$$ for all $$x in an $$interval. What does this tell you about $$f(x)$$?

Accepted Answer

$$f(x) is $$decreasing.

Question 6

A company’s profit function is $$P(x) = -2x^2$ + 8x + 10$$, where $$x is $$the number of units sold. At what value of $$x is $$the profit maximized?

Accepted Answer

$$x = 2$$

Question 7

Given the graph in $$image_1$$, on what interval is the function increasing?

Accepted Answer

$$(-\infty, \infty)$$

Question 8

Given the graph in $$image_2$$, on what interval is the function decreasing?

Accepted Answer

$$(0, 2)$$

Question 9

Given the graph in $$image_3$$, on what interval is the function increasing?

Accepted Answer

$$(0, 1)$$

Question 10

Given the graph in $$image_4$$, on what intervals is the function decreasing?

Accepted Answer

$$(-\infty, -2)$$ and $$(-1, 0)$$

Question 11

For the function $$f(x) = x^4$ + 6x^2$ + 9x - 1$$, what is the first step to find intervals where $$f is $$increasing or decreasing?

Accepted Answer

Find the domain of $$f.$$

Question 12

If $$f'(x) = 0$$ for all $$x in an $$interval, what can you conclude about $$f(x) on $$that interval?

Accepted Answer

$$f(x) is $$constant.

Question 13

A function $$f$$ has a critical number at $$x = 3. $$Which of the following could be true?

Accepted Answer

$$f'(3) = 0$$

Question 14

Given the graph in $$image_6$$, at what $$x-$$value does the function have a critical point?

Accepted Answer

$$x = 0$$

Question 15

A function $$f is $$defined and differentiable everywhere. If $$f'(x)$$ changes from positive to negative at $$x = a$$, what does $$f$$ have at $$x = a$$?

Accepted Answer

A local maximum

Question 16

Suppose $$f(x)$$ models the temperature (in °C) of a chemical reaction over time $$t (in $$minutes), and $$f'(t) \u003c 0$$ for $$t in$$ $$(10, 20). $$What does this mean?

Accepted Answer

The temperature is decreasing between 10 and 20 minutes.

Increasing and Decreasing Functions: Critical Numbers and Intervals

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