Q: A function $$f(x) is $$increasing on an interval if:

$$f'(x) \u003e 0$$ \text{ on the interval}$

Question 1

Given the function $$f(x) = 3x^2$ - 6x + 2$$, for which value of $$x$$ does the graph have a horizontal tangent line?

Accepted Answer

$$x = 1$$

Question 2

For the function $$f(x) = 5 \ln x - x$$, what is the domain of $$f(x)$$?

Accepted Answer

$$(0, \infty)$$

Question 3

A critical number of a function $$f(x) is $$defined as:

Accepted Answer

Any value in the domain of $$f$$ where $$f'(x) = 0 or$$ $$f'(x)$$ does not exist

Question 4

If $$f'(x)$$ changes from negative to positive at $$x = c$$, what does this indicate about $$f(x) at$$ $$x = c$$?

Accepted Answer

$$f(x)$$ has a local minimum at $$x = c$$

Question 5

For the quadratic function $$f(x) = ax^2$ + bx + c$$, the discriminant is $$D = b^2$ - 4ac. $$What does $$D \u003c 0$$ indicate?

Accepted Answer

The function has no real zeros

Question 6

Suppose the total revenue function for selling $$x$$ desks is $$R(x)$$, and the marginal revenue $$R'(x) is $$decreasing with an $$x-$$intercept at $$x = 400. $$What does this imply about the revenue at $$x = 400$$?

Accepted Answer

Revenue is maximized at $$x = 400$$

Question 7

Given $$f(x) = 3x^2$ - 6x + 2$$, find the intervals where $$f(x) is $$increasing.

Accepted Answer

$$(1, \infty)$$

Question 8

If $$f'(x)$$ changes from positive to negative at $$x = c$$, what does this indicate about $$f(x) at$$ $$x = c$$?

Accepted Answer

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

Question 9

For $$f(x) = 5 \ln x - x$$, find the critical number(s) of $$f(x).$$

Accepted Answer

$$x = 5$$

Question 10

A function $$f(x) is $$increasing on an interval if:

Accepted Answer

$$f'(x) \u003e 0$$ 	ext{ on the interval}$

Question 11

Suppose $$f(x)$$ has critical values at $$x = 2$$ and $$x = 4. If$$ $$f'(x)$$ changes from positive to negative at $$x = 2$$ and from negative to positive at $$x = 4$$, what are the local extrema?

Accepted Answer

Local maximum at $$x = 2$$, local minimum at $$x = 4$$

Question 12

Which of the following statements about critical numbers is correct?

Accepted Answer

Critical numbers must be in the domain of $$f$$

Question 13

If the derivative $$f'(x)$$ does not exist at $$x = c$$, but $$x = c is in $$the domain of $$f$$, what can $$x = c be$$?

Accepted Answer

A critical number

Question 14

For the function $$f(x) = 3x^2$ - 6x + 2$$, what is the value of the discriminant?

Accepted Answer

$$(-6)^2 - 4$$ \cdot 3 \cdot 2 = 36 - 24 = 12$$

Question 15

A minimum of a continuous function occurs when:

Accepted Answer

The graph changes from falling to rising

Question 16

If $$f(x) is $$decreasing on the interval $$(a, b)$$, what can be said about $$f'(x) on $$that interval?

Accepted Answer

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

Question 17

Suppose $$R(x) is $$the total revenue from selling $$x$$ units, and $$R'(x) is $$the marginal revenue. If $$R'(x)$$ crosses zero at $$x = 400$$, what does this mean for $$R(x)$$?

Accepted Answer

$$R(x)$$ has a local maximum at $$x = 400$$

Question 18

If $$f(x) is $$continuous and $$f'(x)$$ changes sign at $$x = c$$, what can be said about $$x = c$$?

Accepted Answer

$$x = c is a $$critical number and a local extremum$

Question 19

For $$f(x) = 5 \ln x - x$$, on which interval is $$f(x)$$ decreasing?

Accepted Answer

$$(5, \infty)$$

First Derivatives and Graphs: Extrema, Critical Points, and Applications

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