Question 1

Suppose $$f(x) is a $$continuous and differentiable function on the interval $$(a, b). If$$ $$f'(x) \u003e 0$$ for all $$x in$$ $$(a, b)$$, which of the following statements is true about $$f(x) on$$ $$(a, b)$$?

Accepted Answer

$$f(x) is $$increasing on $$(a, b).$$

Question 2

Given the function $$f(x) = x^2$ + x - 1$$, on which interval is the function decreasing?

Accepted Answer

$$(-\infty, -\frac{1}{2})$$

Question 3

A critical number of a function $$f(x) is a $$value $$c$$ such that:

Accepted Answer

$$f'(c) = 0 or$$ $$f'(c) is $$undefined, and $$f(x) is $$defined at $$x = c.$$

Question 4

If $$f'(x) \u003c 0$$ for every $$x in$$ $$(a, b)$$, what can you conclude about $$f(x) on$$ $$(a, b)$$?

Accepted Answer

$$f(x) is $$decreasing on $$(a, b).$$

Question 5

Which of the following best describes a relative maximum of a function $$f(x) at$$ $$x = c$$?

Accepted Answer

$$f(c) \geq f(x)$$ for all $$x$$ near $$c.$$

Question 6

Suppose $$f(x) is $$continuous and differentiable, and $$f'(c) = 0. If$$ $$f'(x)$$ changes from positive to negative at $$x = c$$, what does $$f(c)$$ represent?

Accepted Answer

A relative maximum

Question 7

Let $$f(x) = x^4$ + 18x^2$. Find the x$-coordinate of the relative minimum.

Accepted Answer

$$x = -3$$

Question 8

If $$f''(x) \u003e 0$$ for all $$x in an $$interval $$(a, b)$$, what can you conclude about the graph of $$f(x) on$$ $$(a, b)$$?

Accepted Answer

It is concave upward on $$(a, b).$$

Question 9

Suppose $$f(x) is $$twice differentiable and $$f'(c) = 0. If$$ $$f''(c) \u003c 0$$, what does the second derivative test tell you about $$f(c)$$?

Accepted Answer

$$f(c) is a $$relative maximum.

Question 10

Which of the following is NOT a possible reason for a point $$x = c to be an $$inflection point of $$f(x)$$?

Accepted Answer

$$f''(c) \u003e 0$$ and $$f(x) is $$continuous at $$x = c.$$

Question 11

Given $$f(x) = -x^2$ + x + 2$$, find $$f'(2)$$ and interpret its meaning.

Accepted Answer

$$f'(2) = -3$$, which means the function is decreasing at $$x = 2.$$

Question 12

Suppose $$A(t)$$ represents the amount of water (in thousands of gallons) flowing out of a reservoir each day, $$t$$ days from the beginning of the year. If $$A'(230) = -685$$ and $$A''(230) \u003c 0$$, what does this indicate?

Accepted Answer

The outflow is decreasing and the rate of decrease is accelerating.

Question 13

If $$f'(x) \u003e 0$$ and $$f''(x) \u003c 0$$ for all $$x in an $$interval, what does this mean about $$f(x)$$?

Accepted Answer

$$f(x) is $$increasing at a decreasing rate.

Question 14

A company models its revenue (in thousands of dollars) as $$R(x) = 600x^2$ - \frac{x^3$}{5000}$$, where $$x is $$the amount spent on advertising (in thousands of dollars). The point of diminishing returns occurs where:

Accepted Answer

$$R''(x) = 0$$

Question 15

Which of the following statements about the first and second derivatives is correct?

Accepted Answer

If $$f'(x) \u003e 0$$ and $$f''(x) \u003e 0$$, $$f(x) is $$increasing at an increasing rate.

Question 16

Given the graph in the fourth image, which of the following best describes the behavior of the function at the leftmost local maximum?

Accepted Answer

The function changes from increasing to decreasing.

Question 17

If $$f(x) is $$concave downward at $$x = 4$$, which of the following must be true?

Accepted Answer

$$f''(4) \u003c 0$$

Question 18

Which of the following is the correct interpretation of an inflection point?

Accepted Answer

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

Graphs and the Derivative: Applications in Business Calculus

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