Question 1

Which of the following best describes a local extremum as indicated in the image?

Accepted Answer

A point where the function changes from increasing to decreasing or vice versa.

Question 2

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

Accepted Answer

$$f'(c) = 0 or$$ $$f'(c)$$ does not exist.

Question 3

Which test is commonly used to determine whether a critical point is a local maximum, minimum, or neither?

Accepted Answer

First Derivative Test

Question 4

Suppose $$f(x)$$ has a local maximum at $$x = c. $$Which of the following is most likely true about $$f'(x)$$ near $$x = c$$?

Accepted Answer

$$f'(x)$$ changes from positive to negative at $$x = c$$

Question 5

Given the function $$f(x) = x^3$ - 3x^2$ + 2$$, at which $$x-$$value does a local extremum occur?

Accepted Answer

$$x = 0$$ and $$x = 2$$

Question 6

Which of the following real-world scenarios best illustrates a local minimum?

Accepted Answer

The lowest point of unemployment rate during a decade.

Question 7

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

Accepted Answer

$$f(x) is $$increasing on that interval.

Question 8

A business tracks its profit function $$P(x)$$ and finds $$P'(x) = 0 at$$ $$x = 5$$ and $$P''(5) \u003e 0. $$What does this indicate?

Accepted Answer

There is a local minimum at $$x = 5.$$

Question 9

Which of the following is NOT a critical number for $$f(x) = \frac{1}{x-1}$$?

Accepted Answer

$$x = 1$$

Question 10

If $$f'(x) \u003c 0$$ for all $$x in an $$interval, what does this tell you about $$f(x)$$?

Accepted Answer

$$f(x) is $$decreasing on that interval.

Question 11

A function $$f(x)$$ has a critical point at $$x = 3. If$$ $$f'(x)$$ changes from negative to positive at $$x = 3$$, what does this indicate?

Accepted Answer

There is a local minimum at $$x = 3.$$

Question 12

Which of the following steps is NOT part of finding intervals where a function is increasing or decreasing?

Accepted Answer

Set $$f(x) = 0$$ and solve for $$x.$$

Question 13

Given $$f(x) = 4x^3$ + 3x^2$ - 18x + 1$$, what is the derivative $$f'(x)$$?

Accepted Answer

$$f'(x) = 12x^2$ + 6x - 18$$

Question 14

If a function $$f(x) is $$constant on an interval, what must be true about its derivative on that interval?

Accepted Answer

$$f'(x) = 0$$

Question 15

A chemical reaction's rate is modeled by $$r(t)$$, and $$r'(t)$$ changes from positive to negative at $$t = 2. $$What does this mean?

Accepted Answer

The reaction rate has a local maximum at $$t = 2.$$

Graphical Applications of Derivatives: Increasing/Decreasing Functions and Local Extrema

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