Q: Given $$f(x) = x^3$ - 9x^2$ + 24x - 10$$, 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 simplified form of its second derivative $$y''$$?

Accepted Answer

$$y'' = 42x^5$ \ln(x) + 84x^5$ + 12x^6$

Question 2

Find 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

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

Accepted Answer

$$x  4$$

Question 5

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

Accepted Answer

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

Question 6

A manufacturer’s production function is $$T(x) = -0.25x^4$ + 3x^3$ for 0$$ \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

$$x = 5$$

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

$$250 ft$$

Question 9

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

Accepted Answer

$$x = b, x = d$$

Question 10

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

Accepted Answer

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

Question 11

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

Accepted Answer

Second image

Question 12

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

Accepted Answer

Third image

Question 13

If $$s = 7t^3$ + 5t^2$ + 9t + 2 is $$the distance (in ft) traveled in time $$t (in s$$), what is the acceleration at $$t = 1$$?

Accepted Answer

$$a = 56$$

Question 14

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

Accepted Answer

Extreme Value Theorem

Question 15

For $$f(x) = \frac{1}{3}x^3$ + \frac{7}{2}x^2$ + 12x + 90 on$$ $$[-3, 4]$$, which $$x-$$value gives the absolute minimum?

Accepted Answer

$$x = -3$$

Question 16

If $$f''(x)$$ changes sign at $$x = 2$$ and $$x = 4$$, what are the points of inflection for $$f(x)$$?

Accepted Answer

$$x = 2$$ and $$x = 4$$

Question 17

Which of the following best describes the intervals where $$f(x) is $$concave upward if $$f''(x) \u003e 0 on$$ $$(a, b)$$?

Accepted Answer

$$(a, b)$$

Question 18

If $$f'(x) \u003e 0$$ and $$f''(x) \u003c 0 on$$ $$(a, b)$$, which of the following graphs (twenty-third to twenty-sixth images) best represents $$f(x)$$?

Accepted Answer

Third image

Guidance for Identifying Local Extrema from a Graph

Guida di studio - Domande di pratica

Study guide - Flashcard