Question 1

Given the function $f(x) = 2x^3 - 9x^2 + 12x$, what are the intervals where $f(x)$ is increasing?

Accepted Answer

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

Question 2

For the function $f(x) = \frac{x^2}{x^2 - 1}$, which of the following is a critical number?

Accepted Answer

$x = 0$

Question 3

Suppose $f(x)$ is continuous and $f'(c) = 0$. If $f'(x)$ changes from positive to negative at $x = c$, what does $f(x)$ have at $x = c$?

Accepted Answer

A local maximum

Question 4

Given $f(x) = x^4 + 4x^3$, find all critical numbers.

Accepted Answer

$x = 0$ and $x = -3$

Question 5

For $f(x) = x^4 - 4x^2$, at which $x$-values does the function have inflection points?

Accepted Answer

$x = 0$ and $x = 2$

Question 6

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

Accepted Answer

It is concave downward

Question 7

For $f(x) = x^4 + 4x^3$, use the second derivative test to determine the nature of the critical point at $x = -3$.

Accepted Answer

Local minimum

Question 8

Which of the following statements is TRUE about the function $f(x)$ if $f'(x) > 0$ for all $x$ in an interval $I$?

Accepted Answer

$f(x)$ is increasing on $I$

Question 9

For $f(x) = 2x^3 - 9x^2 + 12x$, what are the critical numbers?

Accepted Answer

$x = 1$ and $x = 2$

Question 10

A function $f(x)$ has $f''(x) = 0$ at $x = a$ and the concavity changes at $x = a$. What is $x = a$ called?

Accepted Answer

An inflection point

Question 11

Given $f(x) = \frac{x^2}{x^2 - 1}$, on which intervals is $f(x)$ increasing?

Accepted Answer

$x \in (-\infty, -1)$ and $x \in (1, \infty)$

Question 12

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

Accepted Answer

There is no local maximum or minimum at $x = c$

Question 13

For $f(x) = x^4 - 4x^2$, on which intervals is $f(x)$ concave upward?

Accepted Answer

$x < 0$ and $x > 2$

Question 14

Suppose $f(x)$ is continuous and differentiable. If $f'(x)$ changes from negative to positive at $x = c$, what does $f(x)$ have at $x = c$?

Accepted Answer

A local minimum

Question 15

For $f(x) = x^4 + 4x^3$, what is the value of $f''(x)$ at $x = -3$?

Accepted Answer

$f''(-3) = 108$

Question 16

Which of the following is NOT a critical number for $f(x) = \frac{x^2}{x^2 - 1}$?

Accepted Answer

$x = 1$

Question 17

If $f''(x) > 0$ on an interval, what does this imply about the graph of $f(x)$?

Accepted Answer

It is concave upward

Question 18

For $f(x) = x^4 - 4x^2$, use the second derivative test to determine the nature of the critical point at $x = 2$.

Accepted Answer

Inflection point

Question 19

A pharmaceutical company models the concentration of a drug in the bloodstream by $C(t) = 2t^3 - 9t^2 + 12t$, where $t$ is time in hours. At what time(s) is the concentration increasing?

Accepted Answer

$t < 1$ and $t > 2$

Applications of Differentiation: Increasing/Decreasing Functions, Local Extrema, and Concavity

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