Q: For $$f(x) = x^3$ - 9x^2$ + 24x - 10$$, what are the $$x-$$values of the local extrema?

$$x = 2$$ \text{ (max)}, x = 4$$ \text{ (min)}$$

Question 1

Given the function $$y = x^7$ \ln(x)$$, what is the second derivative $$y''$$?

Accepted Answer

$$y'' = 42x^5$ \ln(x) + 84x^6$

Question 2

Find the fifth derivative of $$f(x) = x^7$ + 6x^5$ + 17x^3$ + 2x + 1190.$$

Accepted Answer

$$f^{(5)}(x$$) = $$2520x^2 + 720$

Question 3

Evaluate $$\lim_{x 	o 3} \frac{x^4$ - 81}{x - 3}.$$

Accepted Answer

$$108$$

Question 4

For the function $$f(x) = x^3$ - 9x^2$ + 24x - 10$$, on which interval is the function increasing?

Accepted Answer

$$x  4$$

Question 5

For $$f(x) = x^3$ - 9x^2$ + 24x - 10$$, what are the $$x-$$values of the local extrema?

Accepted Answer

$$x = 2$$ 	ext{ (max)}, x = 4$$ 	ext{ (min)}$$

Question 6

A manufacturer models backpack production by $$T(x) = -0.25x^4$ + 3x^3$ for 0$$ \leq x \leq 15$$. At what value of x$ does the point of diminishing returns occur?

Accepted Answer

$$x = 9$$

Question 7

An open box is made from a $$24 	imes 24$$ inch cardboard by cutting squares from each corner and folding up the sides. If the depth must be between 3 and 6 inches, what size square should be cut to maximize the volume?

Accepted Answer

$$x = 4$$

Question 8

Susan wants to build a $$10,000$$ square foot pen along a river, using $$13/	ext{ft}$$ maple fencing on one side and $$5/	ext{ft}$$ birch fencing on the other two. What is the minimum cost for the fencing?

Accepted Answer

$$\2$$,500$

Question 9

Given the function $$s = 7t^3$ + 5t^2$ + 9t + 2$$, what is the acceleration at $$t = 1$$?

Accepted Answer

$$a = 47$$

Question 10

For a function $$f$$ with $$f''(x)$$ sign changes at $$x = 2$$ and $$x = 4$$, which intervals is $$f$$ concave upward?

Accepted Answer

$$(-\infty, 2)$$ and $$(4, \infty)$$

Question 11

Which of the following graphs shows a function $$f$$ that is concave upward on the interval $$(a, b)$$?

Accepted Answer

Image 2

Question 12

Which of the following graphs shows a function $$f$$ that is concave downward on the interval $$(a, b)$$?

Accepted Answer

Image 3

Question 13

Given the graph of $$f(x) ($$see image), at which $$x-$$values does $$f(x)$$ have a local maximum?

Accepted Answer

$$x = b, d$$

Question 14

Given the graph of $$f(x) ($$see image), on which interval is $$f(x)$$ decreasing?

Accepted Answer

$$(a, b)$$, $$(c, d)$$, $$(e, h)$$

Question 15

For the function $$f(x) = x^3$ - 9x + 1 on $$the interval $$(-3, 0)$$, which theorem guarantees the existence of an absolute maximum?

Accepted Answer

Extreme Value Theorem

Question 16

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

Accepted Answer

$$f(x) is $$concave upward

Question 17

If $$f'(x) = 0$$ and $$f''(x) \u003c 0 at$$ $$x = c$$, what does $$f(x)$$ have at $$x = c$$?

Accepted Answer

A local maximum

Question 18

Given $$f(x) = \frac{1}{3}x^3$ + \frac{7}{2}x^2$ + 12x + 90 on$$ $$[-3, 4]$$, what is the absolute maximum value?

Accepted Answer

$$f(4)$$

Question 19

If $$f''(x)$$ changes sign at $$x = a$$, what does $$x = a$$ represent for $$f(x)$$?

Accepted Answer

A point of inflection

Guidance for Identifying Local Extrema from a Graph

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