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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