Question 1

Let $$f(x) be a $$differentiable function on the interval $$(a, b). $$According to the theorem for increasing and decreasing functions, which statement is true if $$f'(x) \u003e 0$$ for all $$x in$$ $$(a, b)$$?

Accepted Answer

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

Question 2

Suppose $$f(x) is $$defined and differentiable everywhere, and $$f'(x) = 0$$ only at $$x = 2. $$Which of the following statements must be true?

Accepted Answer

There is a critical point at $$x = 2.$$

Question 3

Given $$f(x) = x^3$ - 6x^2$ + 9x + 1$$, find the $$x-$$values where $$f(x)$$ has local extrema.

Accepted Answer

$$x = 1$$ and $$x = 3$$

Question 4

For the function $$f(x) = x^4$ + 2x^3$ + 5x$$, what is the $$x-$$value of the local minimum?

Accepted Answer

$$x = -\frac{3}{2}$$

Question 5

If $$f'(x)$$ changes from positive to negative at $$x = c$$, what does this indicate about $$f(x) at$$ $$x = c$$?

Accepted Answer

$$f(x)$$ has a local maximum at $$x = c.$$

Question 6

A company tracks the price of bacon over time, and the rate of change $$B'(t) is $$negative for $$0 \u003c t \u003c 42$$, positive for $$42 \u003c t \u003c 48$$, and negative for $$48 \u003c t \u003c 70. $$What can be said about the price $$B(t) at$$ $$t = 42$$ and $$t = 48$$?

Accepted Answer

$$B(t)$$ has a local minimum at $$t = 42$$ and a local maximum at $$t = 48.$$

Question 7

Which of the following is a critical value of $$f(x)$$?

Accepted Answer

Any $$x$$ where $$f'(x) = 0 or$$ $$f'(x)$$ does not exist

Question 8

Given $$f(x) is $$continuous and differentiable, and $$f'(x) \u003c 0$$ for $$x in$$ $$(a, b)$$, what does this imply about $$f(x) on$$ $$(a, b)$$?

Accepted Answer

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

Question 9

If $$f'(x)$$ does not exist at $$x = c$$, which of the following statements is true?

Accepted Answer

$$x = c is a $$critical value.

Question 10

Consider the function $$f(x)$$ with $$f'(x) = 3x^2$ - 12x + 9. $$What are the critical points of $$f(x)$$?

Accepted Answer

$$x = 1$$ and $$x = 3$$

Question 11

A function $$f(x)$$ has $$f'(x) = 0 at$$ $$x = a$$ and $$f'(x)$$ changes sign from negative to positive at $$x = a. $$What does this mean for $$f(x) at$$ $$x = a$$?

Accepted Answer

$$f(x)$$ has a local minimum at $$x = a.$$

Question 12

Which of the following intervals is $$f(x)$$ increasing if $$f'(x) \u003e 0 on$$ $$(a, b)$$ and $$f'(x) \u003c 0 on$$ $$(b, c)$$?

Accepted Answer

$$(a, b)$$

Question 13

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

Accepted Answer

$$f'(c) = 0$$

Question 14

Suppose $$f(x) is $$continuous on $$(-\infty, \infty)$$ and $$f'(x) is $$positive for $$x  0. $$What type of extremum does $$f(x)$$ have at $$x = 0$$?

Accepted Answer

Local maximum

Question 15

A function $$f(x) is $$defined on $$(-\infty, \infty)$$ and $$f'(x)$$ does not exist at $$x = 2. $$Which of the following is true?

Accepted Answer

$$x = 2 is a $$critical value.

Question 16

Given $$f(x) is $$continuous and differentiable, and $$f'(x) = 0 at$$ $$x = a$$, but $$f'(x)$$ does not change sign at $$x = a. $$What does this mean?

Accepted Answer

$$f(x)$$ has no local extremum at $$x = a.$$

Question 17

If $$f'(x) is $$positive for $$x in$$ $$(a, b)$$ and negative for $$x in$$ $$(b, c)$$, what can be said about $$f(x) at$$ $$x = b$$?

Accepted Answer

$$f(x)$$ has a local maximum at $$x = b.$$

Question 18

A business wants to maximize profit modeled by $$P(x)$$, where $$x is $$the number of units sold. If $$P'(x) = 0 at$$ $$x = 100$$ and $$P'(x)$$ changes from positive to negative at $$x = 100$$, what does this mean?

Accepted Answer

Profit is maximized at $$x = 100.$$

Increasing/Decreasing Functions and Local Extrema in Business Calculus

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