Question 1

Given the function $$f(x) = x^3$ - 6x^2$ + 2x + 5$$, on which interval is the graph of $$f$$ concave upward?

Accepted Answer

$$1 \u003c x \u003c 2$$

Question 2

For the function $$f(x) = -x^4$ + 14x^3$ + x + 2$$, what is the $$x-$$coordinate of the inflection point?

Accepted Answer

$$x = 10.5$$

Question 3

If $$f''(x) \u003c 0 on an $$interval, what can you conclude about the graph of $$f on $$that interval?

Accepted Answer

The graph is concave downward.

Question 4

A company’s sales function $$S(x) is $$such that $$S''(x) = 0 at$$ $$x = 20. $$What does $$x = 20$$ represent in terms of sales and advertising?

Accepted Answer

The point of diminishing returns.

Question 5

Given $$T(x) = -0.25x^4$ + 5x^3$ for 0$$ \leq x \leq 15$$, at what value of x$ does the rate of change of T-shirt production reach its maximum?

Accepted Answer

$$x = 10$$

Question 6

For the function $$f(x) = x^3$ + 27x^2$, what are the x$-coordinates of the inflection points?

Accepted Answer

$$x = -18$$

Question 7

Which of the following best describes an inflection point?

Accepted Answer

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

Question 8

Refer to the first image. On which interval is the graph concave downward?

Accepted Answer

From $$b to$$ $$d$$

Question 9

Refer to the second image. At which labeled point does the graph have an inflection point?

Accepted Answer

$$e$$

Question 10

If $$f''(x) \u003e 0$$ for all $$x in an $$interval, which of the following must be true?

Accepted Answer

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

Question 11

A baseball cap manufacturer uses $$T(x) = -0.25x^4$ + 6x^3$ to model production. For which value of x$ does the point of diminishing returns occur?

Accepted Answer

$$x = 12$$

Question 12

If $$f'(x) is $$increasing on an interval, what can you conclude about $$f(x) on $$that interval?

Accepted Answer

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

Question 13

Suppose $$f''(x)$$ changes from positive to negative at $$x = 5. $$What does this indicate about the graph of $$f at$$ $$x = 5$$?

Accepted Answer

There is an inflection point.

Question 14

For $$f(x) = -x^3$ + 3x^2$ + 5x - 4$$, what is the $$x-$$coordinate of the inflection point?

Accepted Answer

$$x = 2$$

Question 15

A company finds that the rate of change of sales is decreasing after spending $$x = 30$$ thousand dollars on advertising. What does this mean?

Accepted Answer

Sales are increasing at a decreasing rate.

Second Derivative Test, Concavity, and Inflection Points in Business Calculus

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