**8.** a. The graph of the *derivative* of a function is given. From this graph, determine the intervals in which the function increases and decreases. Justify your answer. *Graph Description:* - The graph of the derivative \( f'(x) \) displays several peaks and troughs. - \( x \)-axis ranges from \(-5\) to \(5\). - Without specific values, identify where the graph is above or below the \( x \)-axis to assess intervals of increase (where \( f'(x) > 0 \)) and decrease (where \( f'(x) < 0 \)). b. The graph of the *2nd derivative* of a function is given. From this graph, determine the intervals in which the function is concave up and concave down. Justify your answer. *Graph Description:* - The 2nd derivative graph \( f''(x) \) shares similar markings along the \( x \)-axis. - Identify where the graph is above the \( x \)-axis (concave up, \( f''(x) > 0 \)) and below the \( x \)-axis (concave down, \( f''(x) < 0 \)). **9.** Consider the function \( h(x) = 14 + 4x^3 - x^4 \). a. Identify the critical points of the function. b. Determine the intervals on which the function increases and decreases. c. Classify the critical points as relative maximums, relative minimums or neither. d. Determine the intervals on which the function is concave up and concave down. e. Determine the inflection points of the function. f. Use the information from steps (a) – (e) to sketch the graph of the function. **10.** We want to construct a cylindrical can with a bottom but no top that will have a volume of 65 cubic inches. Determine the dimensions of the can that will minimize the amount of material needed to construct the can. **11.** Use Newton’s Method to find the root of the given equation, accurate to three decimal places, that lies in the given interval: \( 2x^3 - 9x^2 + 17x + 20 = 0 \) on \([-1,1]\). **12.** Evaluate the following integrals: a.
**8.** a. The graph of the *derivative* of a function is given. From this graph, determine the intervals in which the function increases and decreases. Justify your answer. *Graph Description:* - The graph of the derivative \( f'(x) \) displays several peaks and troughs. - \( x \)-axis ranges from \(-5\) to \(5\). - Without specific values, identify where the graph is above or below the \( x \)-axis to assess intervals of increase (where \( f'(x) > 0 \)) and decrease (where \( f'(x) < 0 \)). b. The graph of the *2nd derivative* of a function is given. From this graph, determine the intervals in which the function is concave up and concave down. Justify your answer. *Graph Description:* - The 2nd derivative graph \( f''(x) \) shares similar markings along the \( x \)-axis. - Identify where the graph is above the \( x \)-axis (concave up, \( f''(x) > 0 \)) and below the \( x \)-axis (concave down, \( f''(x) < 0 \)). **9.** Consider the function \( h(x) = 14 + 4x^3 - x^4 \). a. Identify the critical points of the function. b. Determine the intervals on which the function increases and decreases. c. Classify the critical points as relative maximums, relative minimums or neither. d. Determine the intervals on which the function is concave up and concave down. e. Determine the inflection points of the function. f. Use the information from steps (a) – (e) to sketch the graph of the function. **10.** We want to construct a cylindrical can with a bottom but no top that will have a volume of 65 cubic inches. Determine the dimensions of the can that will minimize the amount of material needed to construct the can. **11.** Use Newton’s Method to find the root of the given equation, accurate to three decimal places, that lies in the given interval: \( 2x^3 - 9x^2 + 17x + 20 = 0 \) on \([-1,1]\). **12.** Evaluate the following integrals: a.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Please answer question number 13.
![**8.**
a. The graph of the *derivative* of a function is given. From this graph, determine the intervals in which the function increases and decreases. Justify your answer.
*Graph Description:*
- The graph of the derivative \( f'(x) \) displays several peaks and troughs.
- \( x \)-axis ranges from \(-5\) to \(5\).
- Without specific values, identify where the graph is above or below the \( x \)-axis to assess intervals of increase (where \( f'(x) > 0 \)) and decrease (where \( f'(x) < 0 \)).
b. The graph of the *2nd derivative* of a function is given. From this graph, determine the intervals in which the function is concave up and concave down. Justify your answer.
*Graph Description:*
- The 2nd derivative graph \( f''(x) \) shares similar markings along the \( x \)-axis.
- Identify where the graph is above the \( x \)-axis (concave up, \( f''(x) > 0 \)) and below the \( x \)-axis (concave down, \( f''(x) < 0 \)).
**9.** Consider the function \( h(x) = 14 + 4x^3 - x^4 \).
a. Identify the critical points of the function.
b. Determine the intervals on which the function increases and decreases.
c. Classify the critical points as relative maximums, relative minimums or neither.
d. Determine the intervals on which the function is concave up and concave down.
e. Determine the inflection points of the function.
f. Use the information from steps (a) – (e) to sketch the graph of the function.
**10.** We want to construct a cylindrical can with a bottom but no top that will have a volume of 65 cubic inches. Determine the dimensions of the can that will minimize the amount of material needed to construct the can.
**11.** Use Newton’s Method to find the root of the given equation, accurate to three decimal places, that lies in the given interval: \( 2x^3 - 9x^2 + 17x + 20 = 0 \) on \([-1,1]\).
**12.** Evaluate the following integrals:
a.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe51b0353-ef18-40e6-8113-76f2f632483e%2F65b56a18-9fb4-4269-b7aa-6f1950361ec2%2F9dau0tg.jpeg&w=3840&q=75)
Transcribed Image Text:**8.**
a. The graph of the *derivative* of a function is given. From this graph, determine the intervals in which the function increases and decreases. Justify your answer.
*Graph Description:*
- The graph of the derivative \( f'(x) \) displays several peaks and troughs.
- \( x \)-axis ranges from \(-5\) to \(5\).
- Without specific values, identify where the graph is above or below the \( x \)-axis to assess intervals of increase (where \( f'(x) > 0 \)) and decrease (where \( f'(x) < 0 \)).
b. The graph of the *2nd derivative* of a function is given. From this graph, determine the intervals in which the function is concave up and concave down. Justify your answer.
*Graph Description:*
- The 2nd derivative graph \( f''(x) \) shares similar markings along the \( x \)-axis.
- Identify where the graph is above the \( x \)-axis (concave up, \( f''(x) > 0 \)) and below the \( x \)-axis (concave down, \( f''(x) < 0 \)).
**9.** Consider the function \( h(x) = 14 + 4x^3 - x^4 \).
a. Identify the critical points of the function.
b. Determine the intervals on which the function increases and decreases.
c. Classify the critical points as relative maximums, relative minimums or neither.
d. Determine the intervals on which the function is concave up and concave down.
e. Determine the inflection points of the function.
f. Use the information from steps (a) – (e) to sketch the graph of the function.
**10.** We want to construct a cylindrical can with a bottom but no top that will have a volume of 65 cubic inches. Determine the dimensions of the can that will minimize the amount of material needed to construct the can.
**11.** Use Newton’s Method to find the root of the given equation, accurate to three decimal places, that lies in the given interval: \( 2x^3 - 9x^2 + 17x + 20 = 0 \) on \([-1,1]\).
**12.** Evaluate the following integrals:
a.
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