Q: For the function $g(x) = 10x^3 - 6x^4$, find all points of inflection.

$x = 0$ and $x = rac{1}{ ext{sqrt}(2)}$

Q: If $f''(x)$ changes sign at $x = a$, what does this indicate about the graph of $f$ at $x = a$?

$x = a$ is a point of inflection.

Q: A business models its revenue with $R(x) = 3x^2 + 6x - 9$. Find the critical points and determine if they are maxima, minima, or neither.

No critical points; the function is a parabola opening upwards.

Q: Which of the following best describes the concavity of $f(x)$ if $f''(x) < 0$ for all $x$ in an interval $I$?

The graph of $f$ is concave down on $I$.

Q: A company’s cost function is $C(x) = x^3 - 6x^2 + 9x + 2$. Find the $x$-values of all relative extrema and classify each using the Second Derivative Test.

$x = 1$ (maximum), $x = 3$ (minimum)

Q: For the function $f(x) = x^4 - 4x^2$, find all points of inflection.

$x = - ext{sqrt}(2)$ and $x = ext{sqrt}(2)$

Question 1

Suppose $f(x)$ is a twice-differentiable function on an interval $I$. If $f''(x) > 0$ for all $x$ in $I$, which of the following statements is true about the graph of $f$ on $I$?

Accepted Answer

The graph of $f$ is concave up on $I$.

Question 2

Given $f(x) = x^3 + 3x^2 - 9x - 13$, find all critical points and classify each as a relative maximum, minimum, or neither using the Second Derivative Test.

Accepted Answer

Critical points at $x = -3$ (relative maximum) and $x = 1$ (relative minimum).

Question 3

A company models its profit with $k(x) = 2e^x - e^{2x}$. Find the $x$-coordinate of the relative minimum and the corresponding minimum profit.

Accepted Answer

$x = 	ext{ln}(0.5)$, $k(x) = -0.25$

Question 4

Which of the following is the correct statement of the Second Derivative Test for relative extrema?

Accepted Answer

If $f'(c) = 0$ and $f''(c) > 0$, then $f(c)$ is a relative minimum; if $f''(c) < 0$, then $f(c)$ is a relative maximum.

Question 5

For the function $g(x) = 10x^3 - 6x^4$, find all points of inflection.

Accepted Answer

$x = 0$ and $x = rac{1}{	ext{sqrt}(2)}$

Question 6

A pharmaceutical company models the concentration of a drug in the bloodstream by $h(x) = e^x - e^{2x}$. At what $x$-value does the point of inflection occur?

Accepted Answer

$x = 	ext{ln}(0.25)$

Question 7

If $f''(x)$ changes sign at $x = a$, what does this indicate about the graph of $f$ at $x = a$?

Accepted Answer

$x = a$ is a point of inflection.

Question 8

Consider the function $f(x) = (2x-5)^3 + 1$. What is the $x$-coordinate of the point where $f$ is not differentiable, and what does this point represent?

Accepted Answer

$x = 2.5$, point of inflection

Question 9

A business models its revenue with $R(x) = 3x^2 + 6x - 9$. Find the critical points and determine if they are maxima, minima, or neither.

Accepted Answer

No critical points; the function is a parabola opening upwards.

Question 10

Which of the following best describes the concavity of $f(x)$ if $f''(x) < 0$ for all $x$ in an interval $I$?

Accepted Answer

The graph of $f$ is concave down on $I$.

Question 11

A company’s cost function is $C(x) = x^3 - 6x^2 + 9x + 2$. Find the $x$-values of all relative extrema and classify each using the Second Derivative Test.

Accepted Answer

$x = 1$ (maximum), $x = 3$ (minimum)

Question 12

If $f'(c) = 0$ and $f''(c) = 0$, which test should be used to classify the critical point at $x = c$?

Accepted Answer

First Derivative Test

Question 13

For the function $f(x) = x^4 - 4x^2$, find all points of inflection.

Accepted Answer

$x = -	ext{sqrt}(2)$ and $x = 	ext{sqrt}(2)$

Question 14

A manufacturer’s profit function is $P(x) = -2x^2 + 8x + 5$. Find the $x$-value of the relative maximum and the maximum profit.

Accepted Answer

$x = 2$, $P(2) = 13$

Question 15

Which of the following is NOT a necessary condition for a point of inflection at $x = a$?

Accepted Answer

$f'(a) = 0$

Question 16

A company’s cost function is $C(x) = x^3 - 3x^2 + 2x$. At what $x$-value does the function have a point of inflection?

Accepted Answer

$x = 2$

Question 17

If a function $f$ has a critical point at $x = c$ and $f''(c) < 0$, what does this tell you about $f$ at $x = c$?

Accepted Answer

$f(c)$ is a relative maximum.

Applications of Differentiation: Using Second Derivatives to Find Maximum and Minimum Values and Sketch Graphs

Study Guide - Practice Questions

Study Guide - Flashcards