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

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

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

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

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

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

Q: 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 $x = a$?

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

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 $f'(x) > 0$ 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 $0 < t < 42$, positive for $42 < t < 48$, and negative for $48 < t < 70$. 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 $f'(x) < 0$ 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 $f'(x) > 0$ on $(a, b)$ and $f'(x) < 0$ 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 $x < 0$, zero at $x = 0$, and negative 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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