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

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

Question 1

Find the second derivative of $$y = x^7$ \ln(x)$$ and simplify your answer.

Accepted Answer

$$y'' = 42x^5$ \ln(x) + 84x^5$ + 42x^5$/x^2$

Question 2

What is the fifth derivative of $$f(x) = x^7$ + 6x^5$ + 17x^3$ + 2x + 1190$$?

Accepted Answer

$$f^{(5)}(x$$) = $$5040x^2 + 720$

Question 3

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

Accepted Answer

$$108$$

Question 4

Given $$f(x) = x^3$ - 9x^2$ + 24x - 10$$, on which intervals is $$f(x)$$ increasing?

Accepted Answer

$$x  4$$

Question 5

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

Accepted Answer

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

Question 6

A backpack manufacturer models production by $$T(x) = -0.25x^4$ + 3x^3$ for 0$$ \leq x \leq 15$$. At what value of x$ does the maximum rate of change of backpack production 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 maple fencing at $$13/ft on $$one side and birch at $$5/ft on $$the other two sides. What is the length of the side parallel to the river that minimizes cost?

Accepted Answer

$$x = 200 ft$$

Question 9

Given the graph below (see $$image_1$$), on which intervals is $$f(x)$$ decreasing?

Accepted Answer

From $$a to$$ $$b$$, $$c to$$ $$d$$, $$e to$$ $$f$$, and $$g to$$ $$h$$

Question 10

Given $$s = 7t^3$ + 5t^2$ + 9t + 2$$, find the velocity and acceleration at $$t = 1.$$

Accepted Answer

Velocity: $$v = 29$$, Acceleration: $$a = 62$$

Question 11

Which of the following graphs (see images 7-10) shows a function $$f$$ that is concave upward on the interval $$(a, b)$$?

Accepted Answer

Graph B

Question 12

Which of the following graphs (see images 11-14) shows a function $$f$$ that is concave downward on the interval $$(a, b)$$?

Accepted Answer

Graph C

Question 13

Given $$f(x) = e$$^{3x}$$, what is the second derivative f$$''(x)$$?

Accepted Answer

$$f''(x) = 9e$$^{3x}$$

Question 14

For $$f(x) = x^3$ - 9x + 1 on$$ $$(-3, 0)$$, which theorem guarantees an absolute maximum exists?

Accepted Answer

Extreme Value Theorem

Question 15

If $$f''(x)$$ changes sign at $$x = 2$$ and $$x = 4$$, what can you conclude about $$f(x) at $$these points?

Accepted Answer

They are points of inflection.

Question 16

Given $$f(x) is $$continuous and twice differentiable, and $$f''(x)$$ changes from positive to negative at $$x = 2$$, what does this indicate?

Accepted Answer

$$x = 2 is a $$point of inflection

Question 17

Which of the following best describes the domain for the volume function of the open box problem with a $$24 	imes 24$$ inch cardboard and depth between 3 and 6 inches?

Accepted Answer

$$3 \leq x \leq 6$$

Question 18

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

Accepted Answer

$$f(x) is $$increasing

Question 19

Given the cost function $$C(x) = 13x + 10,000/x$$ for a fencing problem, what value of $$x$$ minimizes the cost?

Accepted Answer

$$x = 200$$

Analyzing Local Extrema from a Graph

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