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 $$f''(c) \u003c 0$$, 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 $$f''(a) \u003e 0$$

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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