Question 1

Which of the following is the correct definition of a critical point for a function $$f$$?

Accepted Answer

A number $$c in $$the domain of $$f$$ where $$f'(c) = 0 or$$ $$f'(c)$$ does not exist.

Question 2

Let $$f(x) = x^4$ - 4x^3$. What are the critical points of f$?

Accepted Answer

$$x = 0$$ 	ext{ and } x = 3$

Question 3

Suppose $$f(x) is $$continuous on $$[a, b]. $$According to the Extreme Value Theorem, where can the absolute maximum or minimum of $$f$$ occur?

Accepted Answer

At a critical point of $$f in$$ $$(a, b) or at $$the endpoints $$a or$$ $$b.$$

Question 4

Find the absolute maximum and minimum values of $$f(x) = x^3$ - 3x^2$ - 9x + 5 on$$ $$[-2, 4].$$

Accepted Answer

Absolute max $$10 at$$ $$x = -1$$; absolute min $$-22 at$$ $$x = 3$$

Question 5

Which of the following functions is guaranteed to have both an absolute maximum and an absolute minimum on the given interval?

Accepted Answer

$$f(x) = \sqrt{x} \cos x on$$ $$[0, 5]$$

Question 6

Suppose $$f(x) = x e$$^{-x}$$ on $$[-1, 2]$$. Using the closed interval method, which of the following steps is NOT required?

Accepted Answer

Find where $$f''(x) = 0 to $$determine concavity.

Question 7

Which of the following is the correct statement of the Mean Value Theorem (MVT)?

Accepted Answer

If $$f is $$continuous on $$[a, b]$$ and differentiable on $$(a, b)$$, then there exists $$c \in (a, b)$$ such that $$f'(c) = \frac{f(b) - f(a)}{b - a}.$$

Question 8

A car travels 120 miles in 2 hours on a highway. According to the Mean Value Theorem, what can you conclude?

Accepted Answer

At some instant, the car's instantaneous speed was exactly 60 mph.

Question 9

Let $$f(x) = x^3$. Which of the following is true about x = 0$?

Accepted Answer

$$x = 0 is a $$critical point but not a local extremum.

Question 10

For $$f(x) = x^4$, is (0$$, 0)$$ an inflection point?

Accepted Answer

No, because $$f''(x)$$ does not change sign at $$x = 0.$$

Question 11

Suppose $$f(x) = x^3$ + x + 1 on$$ $$[0, 2]. $$For which $$c in$$ $$(0, 2)$$ does the Mean Value Theorem guarantee $$f'(c) = \frac{f(2) - f(0)}{2 - 0}$$?

Accepted Answer

$$c = \sqrt{\frac{2}{3}}$$

Question 12

Let $$f(x) = x^4$ - 2x^2$. On which intervals is f$ increasing?

Accepted Answer

$$(-1, 0)$$ and $$(1, \infty)$$

Question 13

If $$f'(c) = 0$$ and $$f''(c) = 0$$, what can you conclude about $$f at$$ $$c$$?

Accepted Answer

Nothing yet: more information is needed.

Question 14

Let $$f(x) = x e$$^{x^2}$$. What is the differential dy$?

Accepted Answer

$$dy = (e$$^{x^2}$$ + 2x^2$ e$$^{x^2}$$) dx$$

Question 15

For $$f(x) = x^3$ - 3x$$, where is $$f$$ increasing?

Accepted Answer

$$(-\infty, -1)$$ and $$(1, \infty)$$

Question 16

Suppose $$f(x) = x^4$ - 2x^2$. Use the Second Derivative Test to classify the critical points. What happens at x = 0$?

Accepted Answer

The test is inconclusive at $$x = 0$$ because $$f''(0) = 0.$$

Question 17

Let $$f(x) = x e^x. $$Where is $$f$$ concave up?

Accepted Answer

$$(-1, \infty)$$

Question 18

A function $$f is $$continuous on $$[a, b]$$ and differentiable on $$(a, b). If$$ $$f(a) = f(b)$$, what does Rolle's Theorem guarantee?

Accepted Answer

There exists $$c \in (a, b)$$ such that $$f'(c) = 0.$$

Question 19

Suppose $$f(x) is $$differentiable and $$f'(x) \u003e 0$$ for all $$x in$$ $$(a, b). $$What can you conclude about $$f on$$ $$(a, b)$$?

Accepted Answer

$$f is $$increasing on $$(a, b).$$

Extrema, The Mean Value Theorem, and Monotonicity in Calculus

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