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...
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:**
   \[ F'(x) = 6x^2 \]
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.
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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