Question 1

Which of the following statements correctly describes when a function $$f(x) is $$increasing on an interval?

Accepted Answer

For any $$x_1$$ \u003c $$x_2 in $$the interval, $$f(x_1$) \u003c f(x_2$).$$

Question 2

Given the function $$f(x) = x^3$ - 6x^2$ + 9x$$, find the critical numbers.

Accepted Answer

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

Question 3

Suppose $$f'(x) \u003c 0$$ for all $$x in an $$interval. What can you conclude about $$f(x) on $$that interval?

Accepted Answer

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

Question 4

A company models its total cost to produce $$x$$ liters of a chemical by $$C(x) = x^3$ - 12x^2$ + 48x + 100. On $$which interval is the cost function decreasing?

Accepted Answer

$$2 \u003c x \u003c 8$$

Question 5

Which of the following is NOT a critical number for the function $$f(x) = \sqrt{9 - x^2$}$$?

Accepted Answer

$$x = 2$$

Question 6

For the function $$f(x) = x^2$ - 4x + 3$$, on which interval is $$f(x)$$ increasing?

Accepted Answer

$$(2, \infty)$$

Question 7

If $$f'(x) = 0$$ for all $$x in an $$interval, what does this imply about $$f(x) on $$that interval?

Accepted Answer

$$f(x) is $$constant.

Question 8

Given $$C(x) = \frac{x}{\sqrt{9 - x^2$}}$$, for which interval is $$C(x)$$ decreasing?

Accepted Answer

$$(-3, 0)$$

Question 9

A function $$f(x)$$ has a critical number at $$x = 2. If$$ $$f'(x) \u003e 0$$ for $$x  2$$, what does $$x = 2$$ represent?

Accepted Answer

A local maximum

Question 10

Which of the following functions is always increasing for all $$x in $$its domain?

Accepted Answer

$$f(x) = e^x$$

Question 11

Suppose $$f(x) is $$continuous and differentiable everywhere. If $$f'(x)$$ changes sign from positive to negative at $$x = a$$, what does this indicate about $$f(x) at$$ $$x = a$$?

Accepted Answer

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

Question 12

For the function $$f(x) = x^2$ - 6x + 8$$, find the intervals where $$f(x) is $$decreasing.

Accepted Answer

$$(-\infty, 3)$$

Question 13

A manufacturer finds that the marginal cost (the derivative of cost) is zero at $$x = 5$$ units. What does this mean about the cost function at $$x = 5$$?

Accepted Answer

The cost is at a local minimum or maximum at $$x = 5$$

Question 14

If $$f(x) is $$decreasing on $$(-\infty, a)$$ and increasing on $$(a, \infty)$$, what does $$x = a$$ represent?

Accepted Answer

A local minimum

Question 15

Given $$f(x) = x^4$ - 4x^2$, find all critical numbers.

Accepted Answer

$$x = 0, 2, -2$$

Question 16

Suppose the cost function $$C(x)$$ for producing $$x$$ units is given by $$C(x) = 2x^2$ - 4x + 5. $$For which interval is the cost increasing?

Accepted Answer

$$(1, \infty)$$

Question 17

Which of the following is true about the function $$f(x) = \ln(x)$$ for $$x \u003e 1$$?

Accepted Answer

$$f(x) is $$increasing.

Question 18

A function $$f(x)$$ has a critical number at $$x = 0$$ and $$f'(x)  0$$ for $$x \u003e 0. $$What does $$x = 0$$ represent?

Accepted Answer

A local minimum

Question 19

For the function $$f(x) = x^3$ - 3x^2$ + 2x$$, which interval is $$f(x)$$ increasing?

Accepted Answer

$$(-\infty, 1)$$ and $$(2, \infty)$$

Increasing and Decreasing Functions: Business Calculus Study Notes

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