Q: Which step is NOT part of the five-step process for finding absolute extrema on a closed interval?

Set the derivative equal to the function.

Q: Which of the following is true about absolute extrema?

Absolute extrema can occur at endpoints or critical points.

Q: If $f(x)$ has a critical point at $x = c$, which of the following must be true?

$f'(c) = 0$ or $f'(c)$ is undefined

Q: Which of the following is a necessary condition for a point to be a local extremum?

$f'(x) = 0$ or $f'(x)$ is undefined

Q: Which of the following is the correct process to find the absolute maximum and minimum of a function on a closed interval?

Find critical points, evaluate at endpoints, compare all values.

Question 1

Given the function $f(x) = 6x^2 - x^3$, which of the following $x$-values is a critical point?

Accepted Answer

$x = 2$

Question 2

For the function $f(x) = \frac{x^2}{x^2 - 2}$, which $x$-value is NOT a critical point?

Accepted Answer

$x = \sqrt{2}$

Question 3

Which of the following statements best describes a local maximum?

Accepted Answer

It is the highest value of the function in a small neighborhood around a point.

Question 4

If $f'(c) = 0$ and $f''(c) < 0$, what does $x = c$ represent for the function $f(x)$?

Accepted Answer

A local maximum

Question 5

Given $y = 4 - x^2$ on the interval $-2 \leq x \leq 1$, what is the absolute maximum value?

Accepted Answer

$y = 4$ at $x = 0$

Question 6

Which step is NOT part of the five-step process for finding absolute extrema on a closed interval?

Accepted Answer

Set the derivative equal to the function.

Question 7

For $f(x) = 6x^2 - x^3$, what is the slope of the tangent line at $x = 0$?

Accepted Answer

$0$

Question 8

On the interval $-3 \leq x \leq 3$, the absolute minimum of the function in the graph is located at which $x$-value?

Accepted Answer

$x = -1$

Question 9

If $f'(x)$ does not exist at $x = a$, what can $x = a$ be classified as?

Accepted Answer

A critical point

Question 10

For $y = \csc(	heta)$ on the interval $-\frac{\pi}{5} \leq 	heta \leq \frac{\pi}{4}$, which value of $	heta$ gives the absolute minimum?

Accepted Answer

$	heta = -\frac{\pi}{5}$

Question 11

Which of the following is true about absolute extrema?

Accepted Answer

Absolute extrema can occur at endpoints or critical points.

Question 12

If $f(x)$ has a critical point at $x = c$, which of the following must be true?

Accepted Answer

$f'(c) = 0$ or $f'(c)$ is undefined

Question 13

Which of the following is a necessary condition for a point to be a local extremum?

Accepted Answer

$f'(x) = 0$ or $f'(x)$ is undefined

Question 14

For the function $f(x) = 6x^2 - x^3$, what is the $y$-value at the critical point $x = 4$?

Accepted Answer

$f(4) = -16$

Question 15

Which of the following is the correct process to find the absolute maximum and minimum of a function on a closed interval?

Accepted Answer

Find critical points, evaluate at endpoints, compare all values.

Question 16

If $f(x)$ has a local minimum at $x = a$, which of the following must be true?

Accepted Answer

$f'(a) = 0$ and $f''(a) > 0$

Question 17

For $f(x) = \frac{x^2}{x^2 - 2}$, which $x$-values are excluded from the domain?

Accepted Answer

$x = \sqrt{2}$ and $x = -\sqrt{2}$

Question 18

Which of the following is NOT a critical point for $f(x) = 6x^2 - x^3$?

Accepted Answer

$x = 3$

Question 19

If a function $f(x)$ has a critical point at $x = c$ and $f''(c) = 0$, what can be concluded?

Accepted Answer

The point could be an inflection point.

Question 20

For $y = 4 - x^2$ on $-2 \leq x \leq 1$, what is the absolute minimum value?

Accepted Answer

$y = 0$ at $x = -2$

Critical Points and Absolute Extrema in Calculus

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