Question 1

Let $$f(x) = -10 - x^2$. On which interval is f(x$$)$$ increasing?

Accepted Answer

$$(-\infty, 0)$$

Question 2

For the function $$f(x) = -x^3$ - 5x$$, what can you conclude about its intervals of increase and decrease?

Accepted Answer

It is always decreasing on $$(-\infty, \infty)$$

Question 3

Suppose $$f'(c) = 0$$ and $$f'(x)$$ changes from positive to negative at $$x = c. $$What does this tell you about $$f at$$ $$x = c$$?

Accepted Answer

$$f$$ has a local maximum at $$x = c$$

Question 4

Given $$f(x) = 2x^3$ + 3x^2$ - 120x + 3 on$$ $$[-5, 7]$$, what is the local minimum value and at which $$x$$ does it occur?

Accepted Answer

Local minimum at $$x = 4$$, $$f(4) = -301$$

Question 5

For $$f(x) = 4x^5$ + 5x^4$ - 40x^3$ + 4 on$$ $$[-4, 3]$$, what is the absolute minimum value?

Accepted Answer

$$f(-4) = -252$$

Question 6

Which condition must be met for a point $$x = c to be an $$inflection point of $$f(x)$$?

Accepted Answer

$$f''(c) = 0 or$$ $$f''(c) is $$undefined, and $$f''(x)$$ changes sign at $$c$$

Question 7

If $$f''(x) \u003e 0 on an $$interval, what can you say about the graph of $$f on $$that interval?

Accepted Answer

$$f is $$concave up

Question 8

A pharmaceutical company models the concentration of a drug in the bloodstream by $$C(t)$$, where $$t is $$time in hours. If $$C'(t)$$ changes from positive to negative at $$t = 5$$, what does this indicate?

Accepted Answer

The concentration reaches a local maximum at $$t = 5$$

Question 9

For $$f(x) = -x^6$ + 6x^5$ - 6x + 5$$, find the intervals where $$f is $$concave up.

Accepted Answer

$$(0, 4)$$

Question 10

Suppose $$f''(c) \u003c 0$$ and $$f'(c) = 0. $$What does the second derivative test tell you about $$f at$$ $$x = c$$?

Accepted Answer

$$f$$ has a local maximum at $$x = c$$

Question 11

A company’s profit function $$P(x)$$ has $$P'(x) = 0 at$$ $$x = 100$$ units and $$P''(100) \u003e 0. $$What does this mean for the profit at $$x = 100$$?

Accepted Answer

There is a local minimum profit at $$x = 100$$

Question 12

If $$f'(x)$$ does not change sign at $$x = c$$ where $$f'(c) = 0$$, what can you conclude?

Accepted Answer

$$x = c is $$not a local extremum

Question 13

For $$f(x) = -x^6$ + 6x^5$ - 6x + 5$$, at which $$x$$ values do inflection points occur?

Accepted Answer

$$x = 0$$ and $$x = 4$$

Question 14

A biologist models the population of a species by $$P(t)$$, where $$t is $$time. If $$P''(t)$$ changes sign at $$t = 8$$, what does this indicate?

Accepted Answer

There is an inflection point at $$t = 8$$

Question 15

Which of the following is true about the relationship between the first and second derivatives and the shape of the graph of $$f(x)$$?

Accepted Answer

If $$f'(x) \u003e 0$$, $$f is $$increasing; if $$f''(x) \u003e 0$$, $$f is $$concave up

Question 16

A function $$f(x)$$ has $$f'(x) = 0 at$$ $$x = 2$$ and $$f''(2) = 0. $$What should you do to determine if $$x = 2 is a $$local extremum?

Accepted Answer

Use the first derivative test

Question 17

If $$f''(x) \u003c 0 on an $$interval, what does this mean for the graph of $$f$$?

Accepted Answer

$$f is $$concave down

Question 18

A chemical reaction rate $$R(t) is $$modeled by a function where $$R''(t) = 0$$ and $$R''(t)$$ changes sign at $$t = 3. $$What does this mean physically?

Accepted Answer

The reaction rate has an inflection point at $$t = 3$$

Question 19

Suppose $$f(x) is $$always decreasing on $$(-\infty, \infty). $$Which of the following must be true?

Accepted Answer

$$f'(x) \u003c 0$$ for all $$x$$

What Derivatives Tell Us: 4.3

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