Question 1

Let $f(x) = -10 - x^2$. On which interval is $f(x)$ increasing?

Accepted Answer

$(-\infty, 0)$

Question 2

For the function $f(x) = -x^3 - 5x$, what can you conclude about its intervals of increase and decrease?

Accepted Answer

It is always decreasing on $(-\infty, \infty)$

Question 3

Suppose $f'(c) = 0$ and $f'(x)$ changes from positive to negative at $x = c$. What does this tell you about $f$ at $x = c$?

Accepted Answer

$f$ has a local maximum at $x = c$

Question 4

Given $f(x) = 2x^3 + 3x^2 - 120x + 3$ on $[-5, 7]$, what is the local minimum value and at which $x$ does it occur?

Accepted Answer

Local minimum at $x = 4$, $f(4) = -301$

Question 5

For $f(x) = 4x^5 + 5x^4 - 40x^3 + 4$ on $[-4, 3]$, what is the absolute minimum value?

Accepted Answer

$f(-4) = -252$

Question 6

Which condition must be met for a point $x = c$ to be an inflection point of $f(x)$?

Accepted Answer

$f''(c) = 0$ or $f''(c)$ is undefined, and $f''(x)$ changes sign at $c$

Question 7

If $f''(x) > 0$ on an interval, what can you say about the graph of $f$ on that interval?

Accepted Answer

$f$ is concave up

Question 8

A pharmaceutical company models the concentration of a drug in the bloodstream by $C(t)$, where $t$ is time in hours. If $C'(t)$ changes from positive to negative at $t = 5$, what does this indicate?

Accepted Answer

The concentration reaches a local maximum at $t = 5$

Question 9

For $f(x) = -x^6 + 6x^5 - 6x + 5$, find the intervals where $f$ is concave up.

Accepted Answer

$(0, 4)$

Question 10

Suppose $f''(c) < 0$ and $f'(c) = 0$. What does the second derivative test tell you about $f$ at $x = c$?

Accepted Answer

$f$ has a local maximum at $x = c$

Question 11

A company’s profit function $P(x)$ has $P'(x) = 0$ at $x = 100$ units and $P''(100) > 0$. What does this mean for the profit at $x = 100$?

Accepted Answer

There is a local minimum profit at $x = 100$

Question 12

If $f'(x)$ does not change sign at $x = c$ where $f'(c) = 0$, what can you conclude?

Accepted Answer

$x = c$ is not a local extremum

Question 13

For $f(x) = -x^6 + 6x^5 - 6x + 5$, at which $x$ values do inflection points occur?

Accepted Answer

$x = 0$ and $x = 4$

Question 14

A biologist models the population of a species by $P(t)$, where $t$ is time. If $P''(t)$ changes sign at $t = 8$, what does this indicate?

Accepted Answer

There is an inflection point at $t = 8$

Question 15

Which of the following is true about the relationship between the first and second derivatives and the shape of the graph of $f(x)$?

Accepted Answer

If $f'(x) > 0$, $f$ is increasing; if $f''(x) > 0$, $f$ is concave up

Question 16

A function $f(x)$ has $f'(x) = 0$ at $x = 2$ and $f''(2) = 0$. What should you do to determine if $x = 2$ is a local extremum?

Accepted Answer

Use the first derivative test

Question 17

If $f''(x) < 0$ on an interval, what does this mean for the graph of $f$?

Accepted Answer

$f$ is concave down

Question 18

A chemical reaction rate $R(t)$ is modeled by a function where $R''(t) = 0$ and $R''(t)$ changes sign at $t = 3$. What does this mean physically?

Accepted Answer

The reaction rate has an inflection point at $t = 3$

Question 19

Suppose $f(x)$ is always decreasing on $(-\infty, \infty)$. Which of the following must be true?

Accepted Answer

$f'(x) < 0$ for all $x$

What Derivatives Tell Us: 4.3

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