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 $$x in an $$interval $$(a, b)$$, what can you conclude about $$f(x) on $$that interval?

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 $$f(x) = 2x^3$ + 3x^2$ - 120x. To $$maximize profit, which calculus concept should be used?

Accepted Answer

Find where $$f'(x) = 0$$ and use the first derivative test

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

Which of the following best describes a local maximum?

Accepted Answer

$$f(x)$$ changes from increasing to decreasing at that point

Question 11

Suppose $$f(x) is $$decreasing on $$(a, b). $$What can you say about $$f'(x) on$$ $$(a, b)$$?

Accepted Answer

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

Question 12

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

Accepted Answer

$$x = 2$$

Question 13

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

Accepted Answer

$$f(2) = 7$$

Question 14

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

Accepted Answer

$$f'(c) = 0$$

Question 15

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

Accepted Answer

$$f(1) = -8$$

Business Calculus Study Guide: First Derivative Test & Local Extrema

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