Q: Which of the following statements about absolute and relative extrema is TRUE?

A function can have at most one absolute maximum and one absolute minimum on a given set.

Q: Let $f(x) = |x|$. Which of the following is TRUE about the critical points of $f$?

$x = 0$ is a critical point because $f'(0)$ does not exist.

Q: Suppose $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$. Which statement best describes the Mean Value Theorem (MVT)?

There exists $c \in (a, b)$ such that $f'(c) = \dfrac{f(b) - f(a)}{b - a}$.

Q: A company’s profit function is $P(x) = -2x^2 + 8x + 5$, where $x$ is the number of units sold. What is the maximum profit, and at how many units sold does it occur?

Maximum profit is $21$ at $x = 2$ units.

Q: If $f(x)$ is continuous on $[0, 2]$ and $f(0) = 1$, $f(2) = 5$, which of the following must be true by the Mean Value Theorem?

There exists $c \in (0, 2)$ such that $f'(c) = 2$

Q: A function $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$. If $f'(x) = 0$ for all $x \in (a, b)$, what can you conclude about $f$?

$f$ is a constant function on $[a, b]$

Q: Suppose $f(x)$ is continuous on $[a, b]$ and differentiable on $(a, b)$. If $f(a) < f(b)$, what does the Mean Value Theorem guarantee?

There exists $c \in (a, b)$ such that $f'(c) = \dfrac{f(b) - f(a)}{b - a} > 0$

Question 1

Which of the following is the correct set notation for the half-open interval from $a$ to $b$ (including $a$ but not $b$)?

Accepted Answer

$\{ x \in \mathbb{R} \mid a \leq x < b \}$

Question 2

Suppose $f(x) = x^3 - x$ for all $x \in \mathbb{R}$. On which of the following intervals is $f(x)$ negative?

Accepted Answer

$(-\infty, -1)$

Question 3

A function $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$. If $f(a) = f(b)$, which theorem guarantees that there is at least one $c \in (a, b)$ such that $f'(c) = 0$?

Accepted Answer

Rolle's Theorem

Question 4

Which of the following statements about absolute and relative extrema is TRUE?

Accepted Answer

A function can have at most one absolute maximum and one absolute minimum on a given set.

Question 5

Let $f(x) = |x|$. Which of the following is TRUE about the critical points of $f$?

Accepted Answer

$x = 0$ is a critical point because $f'(0)$ does not exist.

Question 6

Suppose $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$. Which statement best describes the Mean Value Theorem (MVT)?

Accepted Answer

There exists $c \in (a, b)$ such that $f'(c) = \dfrac{f(b) - f(a)}{b - a}$.

Question 7

A company’s profit function is $P(x) = -2x^2 + 8x + 5$, where $x$ is the number of units sold. What is the maximum profit, and at how many units sold does it occur?

Accepted Answer

Maximum profit is $21$ at $x = 2$ units.

Question 8

If $f(x)$ is continuous on $[0, 2]$ and $f(0) = 1$, $f(2) = 5$, which of the following must be true by the Mean Value Theorem?

Accepted Answer

There exists $c \in (0, 2)$ such that $f'(c) = 2$

Question 9

Which of the following intervals is unbounded?

Accepted Answer

$(-\infty, 5)$

Question 10

Suppose $f(x)$ is continuous everywhere and $f(x) = x^3 - x$. Which of the following is a correct list of all real zeros of $f(x)$?

Accepted Answer

$x = -1, 0, 1$

Question 11

A function $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$. If $f'(x) = 0$ for all $x \in (a, b)$, what can you conclude about $f$?

Accepted Answer

$f$ is a constant function on $[a, b]$

Question 12

Which of the following is NOT a requirement for applying Rolle's Theorem to a function $f$ on $[a, b]$?

Accepted Answer

$f$ is increasing on $[a, b]$

Question 13

Suppose $f(x)$ is continuous on $[a, b]$ and differentiable on $(a, b)$. If $f(a) < f(b)$, what does the Mean Value Theorem guarantee?

Accepted Answer

There exists $c \in (a, b)$ such that $f'(c) = \dfrac{f(b) - f(a)}{b - a} > 0$

Question 14

If $f(x)$ has a local maximum at $x = c$ and $f$ is differentiable at $c$, what must be true?

Accepted Answer

$f'(c) = 0$

Question 15

Which of the following is a critical number of $f(x) = x^3$?

Accepted Answer

$x = 0$

Question 16

Suppose $f(x)$ is continuous on $[a, b]$ and attains its absolute maximum at $x = c$ in $[a, b]$. Which of the following must be true?

Accepted Answer

$f(c) \geq f(x)$ for all $x \in [a, b]$

Question 17

A function $f$ is continuous on $[0, 3]$ and differentiable on $(0, 3)$. If $f(0) = 2$ and $f(3) = 2$, which theorem guarantees that there is some $c \in (0, 3)$ with $f'(c) = 0$?

Accepted Answer

Rolle's Theorem

Question 18

Suppose $f(x)$ is continuous on $[a, b]$ and differentiable on $(a, b)$. If $f'(x) > 0$ for all $x \in (a, b)$, what can you conclude about $f$ on $[a, b]$?

Accepted Answer

$f$ is increasing on $[a, b]$

(L14) Intervals, Extrema, and the Mean Value Theorem in Business Calculus

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