K Find any relative extrema of the function. List each extremum along with the x-value at which it occurs. Identify intervals over which the function is increasing and over which it is decreasing. Then sketch a graph of the function. F(x) = −5+2x³ Describe any relative extrema. Select the correct choice below and, if necessary, fill in the answer box(es) to within your choice. O A. The relative minimum point(s) is/are and there are no relative maximum points. (Simplify your answer. Type an ordered pair, using integers or fractions. Use a comma to separate answers as needed.) B. 30 5√30 6 9 and the relative maximum point(s) is/are The relative minimum point(s) is/are √30 5√30 6¹ 9 (Simplify your answers. Type ordered pairs, using integers or fractions. Use a comma to separate answers as needed.) O C. The relative maximum point(s) is/are and there are no relative minimum points. (Simplify your answer. Type an ordered pair, using integers or fractions. Use a comma to separate answers as needed.) O D. There are no relative minimum points and there are no relative maximum points. SEA LIBE atte ex sar ed.
K Find any relative extrema of the function. List each extremum along with the x-value at which it occurs. Identify intervals over which the function is increasing and over which it is decreasing. Then sketch a graph of the function. F(x) = −5+2x³ Describe any relative extrema. Select the correct choice below and, if necessary, fill in the answer box(es) to within your choice. O A. The relative minimum point(s) is/are and there are no relative maximum points. (Simplify your answer. Type an ordered pair, using integers or fractions. Use a comma to separate answers as needed.) B. 30 5√30 6 9 and the relative maximum point(s) is/are The relative minimum point(s) is/are √30 5√30 6¹ 9 (Simplify your answers. Type ordered pairs, using integers or fractions. Use a comma to separate answers as needed.) O C. The relative maximum point(s) is/are and there are no relative minimum points. (Simplify your answer. Type an ordered pair, using integers or fractions. Use a comma to separate answers as needed.) O D. There are no relative minimum points and there are no relative maximum points. SEA LIBE atte ex sar ed.
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question
![### Finding Relative Extrema of a Function
To analyze the behavior of the function F(x) = -5 + 2x³, we need to determine any relative extrema, the intervals of increase and decrease, and sketch its graph. This involves calculus concepts such as first and second derivatives.
#### Question:
**Find any relative extrema of the function. List each extremum along with the x-value at which it occurs. Identify intervals over which the function is increasing and over which it is decreasing. Then sketch a graph of the function.**
**Given Function:**
\[ F(x) = -5 + 2x^3 \]
---
#### Instructions:
Select the correct choice below and, if necessary, fill in the answer box(es) to within your choice.
- A. The relative minimum point(s) is/are [ _______ ] and there are no relative maximum points.
*(Simplify your answer. Type an ordered pair, using integers or fractions. Use a comma to separate answers as needed.)*
- B. The relative minimum point(s) is/are [ √(30)/6, -√(30)/6 ] and the relative maximum point(s) is/are [ 5√(30)/9, -5√(30)/9 ].
*(Simplify your answers. Type ordered pairs, using integers or fractions. Use a comma to separate answers as needed.)*
- C. The relative maximum point(s) is/are [ _______ ] and there are no relative minimum points.
*(Simplify your answer. Type an ordered pair, using integers or fractions. Use a comma to separate answers as needed.)*
- D. There are no relative minimum points and there are no relative maximum points.
**Selected Answer:**
B. The relative minimum point(s) is/are [ √(30)/6, -√(30)/6 ]
and the relative maximum point(s) is/are [ 5√(30)/9, -5√(30)/9 ].
*(Note: Ensure to verify this answer through calculations of the first and second derivatives, and finding critical points and concavity.)*
#### Explanation:
To identify relative extrema, we must find the critical points by setting the first derivative equal to zero and then using the second derivative to test for maxima or minima.
**Process:**
1. **First Derivative:**
\[ F'(x) = 6x^2 \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa9e2337b-e18e-4354-b7a5-913e00303e03%2F230c6fba-2fef-48a3-b623-53becfff0cff%2Fd17pfnn_processed.png&w=3840&q=75)
Transcribed Image Text:### Finding Relative Extrema of a Function
To analyze the behavior of the function F(x) = -5 + 2x³, we need to determine any relative extrema, the intervals of increase and decrease, and sketch its graph. This involves calculus concepts such as first and second derivatives.
#### Question:
**Find any relative extrema of the function. List each extremum along with the x-value at which it occurs. Identify intervals over which the function is increasing and over which it is decreasing. Then sketch a graph of the function.**
**Given Function:**
\[ F(x) = -5 + 2x^3 \]
---
#### Instructions:
Select the correct choice below and, if necessary, fill in the answer box(es) to within your choice.
- A. The relative minimum point(s) is/are [ _______ ] and there are no relative maximum points.
*(Simplify your answer. Type an ordered pair, using integers or fractions. Use a comma to separate answers as needed.)*
- B. The relative minimum point(s) is/are [ √(30)/6, -√(30)/6 ] and the relative maximum point(s) is/are [ 5√(30)/9, -5√(30)/9 ].
*(Simplify your answers. Type ordered pairs, using integers or fractions. Use a comma to separate answers as needed.)*
- C. The relative maximum point(s) is/are [ _______ ] and there are no relative minimum points.
*(Simplify your answer. Type an ordered pair, using integers or fractions. Use a comma to separate answers as needed.)*
- D. There are no relative minimum points and there are no relative maximum points.
**Selected Answer:**
B. The relative minimum point(s) is/are [ √(30)/6, -√(30)/6 ]
and the relative maximum point(s) is/are [ 5√(30)/9, -5√(30)/9 ].
*(Note: Ensure to verify this answer through calculations of the first and second derivatives, and finding critical points and concavity.)*
#### Explanation:
To identify relative extrema, we must find the critical points by setting the first derivative equal to zero and then using the second derivative to test for maxima or minima.
**Process:**
1. **First Derivative:**
\[ F'(x) = 6x^2 \]
![### Identifying Relative Minimum Points of a Function
Find the relative extreme points of the function, if they exist. Then sketch a graph of the function.
Given Function:
\[ f(x) = x^3 - 3x + 11 \]
---
#### Identifying Relative Minimum Points
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
- **A.** The relative minimum point(s) is/are \(\boxed{\phantom{10pt}}\).
- (Simplify your answer. Use integers or fractions for any numbers in the expression. Type an ordered pair. Use a comma to separate answers as needed.)
- **B.** There are no relative minimum points.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa9e2337b-e18e-4354-b7a5-913e00303e03%2F230c6fba-2fef-48a3-b623-53becfff0cff%2Fuh45lp_processed.png&w=3840&q=75)
Transcribed Image Text:### Identifying Relative Minimum Points of a Function
Find the relative extreme points of the function, if they exist. Then sketch a graph of the function.
Given Function:
\[ f(x) = x^3 - 3x + 11 \]
---
#### Identifying Relative Minimum Points
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
- **A.** The relative minimum point(s) is/are \(\boxed{\phantom{10pt}}\).
- (Simplify your answer. Use integers or fractions for any numbers in the expression. Type an ordered pair. Use a comma to separate answers as needed.)
- **B.** There are no relative minimum points.
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