Question 1

Given the function $$f(x) = 2x^3$ - 9x^2$ + 12x$$, what are the intervals where $$f(x) is $$increasing?

Accepted Answer

$$x \in (-\infty, 1)$$ and $$x \in (2, \infty)$$

Question 2

For the function $$f(x) = \frac{x^2$}{x^2$ - 1}$$, which of the following is a critical number?

Accepted Answer

$$x = 0$$

Question 3

Suppose $$f(x) is $$continuous and $$f'(c) = 0. If$$ $$f'(x)$$ changes from positive to negative at $$x = c$$, what does $$f(x)$$ have at $$x = c$$?

Accepted Answer

A local maximum

Question 4

Given $$f(x) = x^4$ + 4x^3$, find all critical numbers.

Accepted Answer

$$x = 0$$ and $$x = -3$$

Question 5

For $$f(x) = x^4$ - 4x^2$, at which x$-values does the function have inflection points?

Accepted Answer

$$x = 0$$ and $$x = 2$$

Question 6

If $$f''(x) \u003c 0 on an $$interval, what can you conclude about the graph of $$f(x) on $$that interval?

Accepted Answer

It is concave downward

Question 7

For $$f(x) = x^4$ + 4x^3$, use the second derivative test to determine the nature of the critical point at x = -3$.

Accepted Answer

Local minimum

Question 8

Which of the following statements is TRUE about the function $$f(x) if$$ $$f'(x) \u003e 0$$ for all $$x in an $$interval $$I$$?

Accepted Answer

$$f(x) is $$increasing on $$I$$

Question 9

For $$f(x) = 2x^3$ - 9x^2$ + 12x$$, what are the critical numbers?

Accepted Answer

$$x = 1$$ and $$x = 2$$

Question 10

A function $$f(x)$$ has $$f''(x) = 0 at$$ $$x = a$$ and the concavity changes at $$x = a. $$What is $$x = a$$ called?

Accepted Answer

An inflection point

Question 11

Given $$f(x) = \frac{x^2$}{x^2$ - 1}$$, on which intervals is $$f(x)$$ increasing?

Accepted Answer

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

Question 12

If $$f'(x)$$ does not change sign at a critical number $$x = c$$, what can you conclude?

Accepted Answer

There is no local maximum or minimum at $$x = c$$

Question 13

For $$f(x) = x^4$ - 4x^2$, on which intervals is f(x$$)$$ concave upward?

Accepted Answer

$$x  2$$

Question 14

Suppose $$f(x) is $$continuous and differentiable. If $$f'(x)$$ changes from negative to positive at $$x = c$$, what does $$f(x)$$ have at $$x = c$$?

Accepted Answer

A local minimum

Question 15

For $$f(x) = x^4$ + 4x^3$, what is the value of f$$''(x)$$ at x = -3$?

Accepted Answer

$$f''(-3) = 108$$

Question 16

Which of the following is NOT a critical number for $$f(x) = \frac{x^2$}{x^2$ - 1}$$?

Accepted Answer

$$x = 1$$

Question 17

If $$f''(x) \u003e 0 on an $$interval, what does this imply about the graph of $$f(x)$$?

Accepted Answer

It is concave upward

Question 18

For $$f(x) = x^4$ - 4x^2$, use the second derivative test to determine the nature of the critical point at x = 2$.

Accepted Answer

Inflection point

Question 19

A pharmaceutical company models the concentration of a drug in the bloodstream by $$C(t) = 2t^3$ - 9t^2$ + 12t$$, where $$t is $$time in hours. At what time(s) is the concentration increasing?

Accepted Answer

$$t  2$$

Applications of Differentiation: Increasing/Decreasing Functions, Local Extrema, and Concavity

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