Question 1

Which of the following statements best describes a function that is concave up at $$x = c$$?

Accepted Answer

The slope of the graph increases as you move left to right through $$(c, f(c)).$$

Question 2

Given a function $$f(x)$$, which condition must be true for $$f to be $$concave down at $$x = c$$?

Accepted Answer

$$f''(c) \u003c 0$$

Question 3

What is an inflection point of a function $$f(x)$$?

Accepted Answer

A point where $$f''(x)$$ changes sign and $$f(x) is $$continuous.

Question 4

Suppose the graph of $$f(x) is $$tangent to its tangent line at $$x = c$$ and lies above the tangent line near $$x = c. $$What can you conclude?

Accepted Answer

The function is concave up at $$x = c.$$

Question 5

If $$f''(x) \u003e 0$$ for all $$x in an $$interval, which of the following is true about $$f(x) on $$that interval?

Accepted Answer

$$f(x) is $$concave up.

Question 6

Consider the function $$f(x)$$ shown in the fourth image. On which interval is the function concave down?

Accepted Answer

$$(4, \infty)$$

Question 7

A company’s sales are modeled by a function $$S(t)$$, where $$t is $$time in months. If $$S''(t) \u003c 0$$ during months 3 to 5, what does this indicate about the sales trend?

Accepted Answer

Sales are increasing at a decreasing rate.

Question 8

Which of the following is true at a relative minimum of a function $$f(x)$$?

Accepted Answer

$$f'(x) = 0$$ and $$f''(x) \u003e 0$$

Question 9

If a function $$f(x) is $$concave up on $$(-\infty, -1) \cup (1, 4)$$, what can you say about $$f''(x) on $$these intervals?

Accepted Answer

$$f''(x) \u003e 0$$

Question 10

Which of the following best describes the behavior of the slope of a function that is concave down?

Accepted Answer

The slope decreases as you move left to right.

Question 11

A function $$f(x)$$ has inflection points at $$x = -1$$ and $$x = 4. $$What happens to the concavity of $$f(x) at $$these points?

Accepted Answer

The function changes from concave up to concave down or vice versa.

Question 12

Suppose $$f(x) is $$continuous and $$f''(x)$$ changes sign at $$x = c. $$What can you conclude?

Accepted Answer

$$x = c is an $$inflection point.

Question 13

If $$f(x) is $$decreasing on the interval $$(-\infty, -3) \cup (1, 1.8)$$, which of the following must be true about $$f'(x) on $$this interval?

Accepted Answer

$$f'(x) \u003c 0$$

Question 14

Which of the following intervals is $$f(x)$$ increasing, based on the graph in the fourth image?

Accepted Answer

$$(-3, -1) \cup (1.8, 6.4)$$

Question 15

A function $$f(x)$$ has a relative maximum at $$x = -1$$ and a relative minimum at $$x = 1.8. $$Which of the following must be true?

Accepted Answer

$$f'(x)$$ changes from positive to negative at $$x = -1$$ and from negative to positive at $$x = 1.8$$

Question 16

If a function $$f(x) is $$concave up on $$(-\infty, -1)$$ and concave down on $$(4, \infty)$$, what can you infer about the second derivative $$f''(x) on $$these intervals?

Accepted Answer

$$f''(x) \u003e 0 on$$ $$(-\infty, -1)$$ and $$f''(x) \u003c 0 on$$ $$(4, \infty)$$

Question 17

Which of the following best describes the relationship between the second derivative and concavity?

Accepted Answer

If $$f''(x) \u003e 0$$, the function is concave up.

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