2. Consider the function f (x) = x¹ - 2x² +3. (a) Find the intervals on which f is increasing / decreasing. (b) Find the intervals on which f is concave up/ concave down. Display your answer in table form, with a column for the intervals, columns for each factor, a column for the derivative, and a column for the function, as shown in lecture.
2. Consider the function f (x) = x¹ - 2x² +3. (a) Find the intervals on which f is increasing / decreasing. (b) Find the intervals on which f is concave up/ concave down. Display your answer in table form, with a column for the intervals, columns for each factor, a column for the derivative, and a column for the function, as shown in lecture.
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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![### Calculus: Analyzing the Function \( f(x) = x^4 - 2x^2 + 3 \)
#### Problem Statement
Consider the function \( f(x) = x^4 - 2x^2 + 3 \).
1. **Find the intervals on which \( f \) is increasing/decreasing.**
2. **Find the intervals on which \( f \) is concave up/concave down.**
#### Instructions
Display your answer in table form, with a column for the intervals, columns for each factor, a column for the derivative, and a column for the function, as shown in lecture.
---
### Solution
First, we need to compute the first derivative of the function to determine the intervals where the function is increasing or decreasing.
Given: \( f(x) = x^4 - 2x^2 + 3 \)
#### Step 1: Compute the First Derivative
\[ f'(x) = \frac{d}{dx} (x^4 - 2x^2 + 3) \]
\[ f'(x) = 4x^3 - 4x \]
#### Step 2: Find Critical Points
Set the first derivative to zero and solve for \( x \):
\[ 4x^3 - 4x = 0 \]
\[ 4x(x^2 - 1) = 0 \]
\[ 4x(x - 1)(x + 1) = 0 \]
The critical points are: \( x = 0, 1, -1 \)
#### Step 3: Interval Test for Increasing/Decreasing
Test the intervals between and beyond the critical points.
#### Step 4: Compute the Second Derivative
\[ f''(x) = \frac{d}{dx} (4x^3 - 4x) \]
\[ f''(x) = 12x^2 - 4 \]
#### Step 5: Find Points of Inflection
Set the second derivative to zero and solve for \( x \):
\[ 12x^2 - 4 = 0 \]
\[ 12x^2 = 4 \]
\[ x^2 = \frac{1}{3} \]
\[ x = \pm \frac{\sqrt{3}}{3} \]
These are potential inflection points.
#### Step 6: Interval](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F52db8ee6-33d1-4aac-bbe4-ebc771de1e49%2Fdd434ea5-840d-406a-9b8a-c8892a6aab2e%2F127jjk9r_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Calculus: Analyzing the Function \( f(x) = x^4 - 2x^2 + 3 \)
#### Problem Statement
Consider the function \( f(x) = x^4 - 2x^2 + 3 \).
1. **Find the intervals on which \( f \) is increasing/decreasing.**
2. **Find the intervals on which \( f \) is concave up/concave down.**
#### Instructions
Display your answer in table form, with a column for the intervals, columns for each factor, a column for the derivative, and a column for the function, as shown in lecture.
---
### Solution
First, we need to compute the first derivative of the function to determine the intervals where the function is increasing or decreasing.
Given: \( f(x) = x^4 - 2x^2 + 3 \)
#### Step 1: Compute the First Derivative
\[ f'(x) = \frac{d}{dx} (x^4 - 2x^2 + 3) \]
\[ f'(x) = 4x^3 - 4x \]
#### Step 2: Find Critical Points
Set the first derivative to zero and solve for \( x \):
\[ 4x^3 - 4x = 0 \]
\[ 4x(x^2 - 1) = 0 \]
\[ 4x(x - 1)(x + 1) = 0 \]
The critical points are: \( x = 0, 1, -1 \)
#### Step 3: Interval Test for Increasing/Decreasing
Test the intervals between and beyond the critical points.
#### Step 4: Compute the Second Derivative
\[ f''(x) = \frac{d}{dx} (4x^3 - 4x) \]
\[ f''(x) = 12x^2 - 4 \]
#### Step 5: Find Points of Inflection
Set the second derivative to zero and solve for \( x \):
\[ 12x^2 - 4 = 0 \]
\[ 12x^2 = 4 \]
\[ x^2 = \frac{1}{3} \]
\[ x = \pm \frac{\sqrt{3}}{3} \]
These are potential inflection points.
#### Step 6: Interval
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