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

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

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

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

Question 1

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

Accepted Answer

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

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 $f'(x) < 0$ 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

$2 < x < 8$

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 $f'(x) > 0$ for $x < 2$ and $f'(x) < 0$ 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 $x > 1$?

Accepted Answer

$f(x)$ is increasing.

Question 18

A function $f(x)$ has a critical number at $x = 0$ and $f'(x) < 0$ for $x < 0$, $f'(x) > 0$ for $x > 0$. 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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