$$f'(x) \u003c 0$$ \\ \text{ for all } x \text{ in } (a, b)$

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 graph in Image 1. On which interval is $$f(x)$$ increasing?

Accepted Answer

From $$b to$$ $$d$$ and from $$e to$$ $$h$$

Question 4

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

Accepted Answer

$$x = 1$$

Question 5

Given $$f(x) = -3x^2$ + 12x - 5$$, what is the nature of the extremum and its $$x-$$coordinate?

Accepted Answer

Local maximum at $$x = 2$$

Question 6

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 7

Refer to Image 2. At which labeled point does the function have a local minimum?

Accepted Answer

At point $$c$$

Question 8

A business models its profit with $$P(x) = 2x^3$ + 3x^2$ - 120x. To $$maximize profit, at which $$x-$$value should the company operate?

Accepted Answer

$$x = 4$$

Question 9

If $$f'(c) = 0$$ and $$f''(c) \u003e 0$$, what does this indicate about $$f(x) at$$ $$x = c$$?

Accepted Answer

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

Question 10

Suppose $$f(x) is $$decreasing on $$(a, b). $$Which of the following must be true?

Accepted Answer

$$f'(x) \u003c 0$$ \ 	ext{ for all } x 	ext{ in } (a, b)$

Question 11

A company’s cost function is $$C(x) = 3x^2$ - 12x + 2. At $$what production level is the cost minimized?

Accepted Answer

$$x = 2$$

Question 12

Given the function $$f(x) = 2x^3$ + 3x^2$ - 120x$$, which of the following is a critical point?

Accepted Answer

$$x = -10$$

Question 13

If $$f(x)$$ has a local maximum at $$x = c$$, which of the following must be true?

Accepted Answer

$$f'(x)$$ changes from positive to negative at $$x = c$$

Question 14

Refer to Image 1. At which labeled point does the function have a local maximum?

Accepted Answer

At point $$d$$

Question 15

If $$f(x) is $$increasing on $$(a, b)$$ and decreasing on $$(b, c)$$, what can you say about $$f(b)$$?

Accepted Answer

$$f(b) is a $$local maximum

Business Calculus: First Derivative Test, Increasing/Decreasing Intervals, and Local Extrema

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