Question 1

Given the function $$f(x) = 3x^2$ - 12x + 2$$, at what value of $$x$$ does $$f(x)$$ attain its local minimum?

Accepted Answer

$$x = 2$$

Question 2

For the function $$f(x) = -2x^2$ - 16x - 25$$, which of the following statements is true about its local extremum?

Accepted Answer

It has a local maximum at $$x = -4$$

Question 3

Consider the function $$f(x) = 5x^2$ - 10x - 3. On $$which interval is $$f(x)$$ decreasing?

Accepted Answer

$$(-\infty, 1)$$

Question 4

Given the graph in Image 1, on which interval is $$f(x)$$ increasing?

Accepted Answer

$$(b, d)$$ and $$(e, h)$$

Question 5

Given the graph in Image 2, at which labeled point does $$f(x)$$ have a local minimum?

Accepted Answer

Point $$c$$

Question 6

For $$f(x) = -3x^2$ + 12x - 5$$, what is the $$x-$$coordinate of the local maximum?

Accepted Answer

$$x = 2$$

Question 7

If $$f'(x) \u003e 0$$ for all $$x in$$ $$(a, b)$$, what can you conclude about $$f(x) on$$ $$(a, b)$$?

Accepted Answer

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

Question 8

Suppose $$f'(x)$$ 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

A company’s profit function is $$P(x) = 2x^3$ + 3x^2$ - 120x. $$Find the critical points by setting $$P'(x) = 0.$$

Accepted Answer

$$x = -10, x = 4$$

Question 10

Given $$f(x) = 3x^2$ - 12x + 2$$, what is the value of the local minimum?

Accepted Answer

$$f(2) = -10$$

Question 11

If $$f'(x) \u003c 0$$ for all $$x in$$ $$(a, b)$$, what does this mean for $$f(x)$$?

Accepted Answer

$$f(x) is $$decreasing on $$(a, b)$$

Question 12

Suppose $$f(x)$$ models the temperature of a chemical reaction over time. If $$f'(t)$$ changes from negative to positive at $$t = 5$$, what does this indicate?

Accepted Answer

The temperature reaches a local minimum at $$t = 5$$

Question 13

Given the graph in Image 1, at which labeled point does $$f(x)$$ have a local maximum?

Accepted Answer

Point $$d$$

Question 14

For $$f(x) = 5x^2$ - 10x - 3$$, what is the $$x-$$coordinate of the vertex (local minimum)?

Accepted Answer

$$x = 1$$

Question 15

A business wants to maximize its profit, modeled by $$P(x) = 2x^3$ + 3x^2$ - 120x. $$Which test should be used to determine if a critical point is a maximum or minimum?

Accepted Answer

First Derivative Test

First Derivative Test: Increasing/Decreasing Functions and Local Extrema

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