Question 1

Which of the following statements best describes the Extreme Value Theorem?

Accepted Answer

A function that is continuous on a closed interval $[a, b]$ has an absolute maximum and absolute minimum.

Question 2

Given $f(x) = 3x^2 + 5x - 2$, which value of $x$ is a critical point?

Accepted Answer

$x = -\frac{5}{6}$

Question 3

For $f(x) = \sqrt{x}$ on $[0,1]$, which of the following is a critical point?

Accepted Answer

$x = 0$

Question 4

Which of the following is NOT necessarily true about critical points?

Accepted Answer

All critical points are local extrema.

Question 5

Find the critical points of $f(x) = \frac{16x}{x^2 + 64}$.

Accepted Answer

$x = \pm 8$

Question 6

A pharmaceutical company wants to maximize the concentration of a drug in the bloodstream, modeled by $C(t) = 3t^2 - 6t + 1$ on $[0,4]$. What is the absolute maximum concentration?

Accepted Answer

$C(4) = 17$

Question 7

Which of the following steps is NOT part of finding absolute extrema of $f(x)$ on $[a, b]$?

Accepted Answer

Evaluate $f(x)$ at all points in $[a, b]$.

Question 8

Given $f(x) = 2x^3 + 3x^2 - 12x + 4$ on $[0,2]$, what is the absolute minimum value?

Accepted Answer

$f(1) = -3$

Question 9

For $h(x) = (x-1)^{1/3}$ on $[-7,9]$, what is the absolute minimum value?

Accepted Answer

$h(1) = 0$

Question 10

A function $f(x)$ is continuous on $[a, b]$ and has critical points at $x = c_1, c_2$. Which values must be checked to find the absolute maximum and minimum?

Accepted Answer

$f(c_1), f(c_2), f(a), f(b)$

Question 11

If $f'(c) = 0$ at $x = c$ but the graph of $f(x)$ is strictly increasing at $c$, what can be concluded?

Accepted Answer

$x = c$ is a critical point but not a local extremum

Question 12

A biologist models population growth with $f(x) = x^3 - 3x^2 + 1$ on $[-\frac{1}{2}, 4]$. What is the absolute minimum population?

Accepted Answer

$f(2) = -3$

Question 13

Which of the following is a necessary condition for a point $x = c$ to be a local extremum of $f(x)$?

Accepted Answer

$f'(c) = 0$ or $f'(c)$ does not exist

Question 14

A chemical reaction rate is modeled by $f(x) = \frac{16x}{x^2 + 64}$. What are the critical points for maximizing or minimizing the rate?

Accepted Answer

$x = \pm 8$

Question 15

Which of the following functions has a critical point at an endpoint of its domain?

Accepted Answer

$f(x) = \sqrt{x}$ on $[0,1]$

Question 16

A function $f(x)$ is continuous on $[a, b]$ and differentiable on $(a, b)$. Which statement is true?

Accepted Answer

It must have an absolute maximum and minimum on $[a, b]$.

Question 17

Which of the following is a plausible reason why $f'(c)$ might not exist at a critical point?

Accepted Answer

The function has a cusp or corner at $x = c$.

Question 18

A medical researcher models the temperature of a patient as $T(t) = 2t^3 + 3t^2 - 12t + 4$ on $[0,2]$. What is the highest temperature recorded?

Accepted Answer

$T(2) = 8$

Question 19

Which of the following is true about local extrema?

Accepted Answer

Local extrema can only occur at critical points.

Maxima and Minima: 4.1

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