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

$$\{ x \in \mathbb{R} \mid a \leq x \u003c b \}$$

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 $$(a, b). If$$ $$f(a) \u003c f(b)$$, what does the Mean Value Theorem guarantee?

Accepted Answer

There exists $$c \in (a, b)$$ such that $$f'(c) = \dfrac{f(b) - f(a)}{b - a} \u003e 0$$

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 $$(a, b). If$$ $$f'(x) \u003e 0$$ 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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