= ²√x = by identifying the domain and any symmetries, finding the derivatives y' and y", finding the critical points and identifying the function's behavior at √x. x20 each one, finding where the curve is increasing and where it is decreasing, finding the points of inflection, determining the concavity of the curve, identifying any asymptotes, and plotting any key points such as intercepts, critical points, and inflection points. Then find coordinates of absolute extreme points, if any. x<0 Graph the function y=
= ²√x = by identifying the domain and any symmetries, finding the derivatives y' and y", finding the critical points and identifying the function's behavior at √x. x20 each one, finding where the curve is increasing and where it is decreasing, finding the points of inflection, determining the concavity of the curve, identifying any asymptotes, and plotting any key points such as intercepts, critical points, and inflection points. Then find coordinates of absolute extreme points, if any. x<0 Graph the function y=
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![**Graph the Function \( y = \frac{7}{4} \sqrt{|x|} \):**
The function is defined piecewise as:
\[
y = \begin{cases}
\frac{7}{4} \sqrt{-x}, & x < 0 \\
\frac{7}{4} \sqrt{x}, & x \geq 0
\end{cases}
\]
### Analysis Steps:
1. **Identify the Domain and Any Symmetries**:
- The domain of the function encompasses all real numbers, \( x \in \mathbb{R} \).
- The function exhibits symmetry about the y-axis due to the absolute value of \( x \).
2. **Find the Derivatives \( y' \) and \( y'' \)**:
- Compute the first derivative \( y' \) to analyze increasing and decreasing intervals.
- Compute the second derivative \( y'' \) to determine concavity and points of inflection.
3. **Find Critical Points**:
- Identify points where the first derivative equals zero or is undefined.
- Determine the nature of each critical point (maximum, minimum, or saddle point).
4. **Determine Function Behavior**:
- Analyze the function's behavior in different intervals defined by critical points.
5. **Investigate Increasing and Decreasing Intervals**:
- Use the sign of \( y' \) to determine where the function is increasing or decreasing.
6. **Find Points of Inflection**:
- Use \( y'' \) to locate points where concavity changes.
7. **Determine Concavity of the Curve**:
- Identify intervals of concave up and concave down using the second derivative.
8. **Identify Asymptotes**:
- Investigate any vertical or horizontal asymptotes.
9. **Plot Key Points**:
- Locate important points such as intercepts (x and y), critical points, and inflection points.
10. **Find Absolute Extreme Points**:
- Determine and provide coordinates for any absolute maximum or minimum points, if they exist.
By following these steps, a comprehensive understanding and accurate graph of the function can be constructed.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F739452bb-bec9-43d8-9372-35dc57efa9d5%2Fdab7db74-d058-4475-8875-f79e25078690%2Fv4awk4_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Graph the Function \( y = \frac{7}{4} \sqrt{|x|} \):**
The function is defined piecewise as:
\[
y = \begin{cases}
\frac{7}{4} \sqrt{-x}, & x < 0 \\
\frac{7}{4} \sqrt{x}, & x \geq 0
\end{cases}
\]
### Analysis Steps:
1. **Identify the Domain and Any Symmetries**:
- The domain of the function encompasses all real numbers, \( x \in \mathbb{R} \).
- The function exhibits symmetry about the y-axis due to the absolute value of \( x \).
2. **Find the Derivatives \( y' \) and \( y'' \)**:
- Compute the first derivative \( y' \) to analyze increasing and decreasing intervals.
- Compute the second derivative \( y'' \) to determine concavity and points of inflection.
3. **Find Critical Points**:
- Identify points where the first derivative equals zero or is undefined.
- Determine the nature of each critical point (maximum, minimum, or saddle point).
4. **Determine Function Behavior**:
- Analyze the function's behavior in different intervals defined by critical points.
5. **Investigate Increasing and Decreasing Intervals**:
- Use the sign of \( y' \) to determine where the function is increasing or decreasing.
6. **Find Points of Inflection**:
- Use \( y'' \) to locate points where concavity changes.
7. **Determine Concavity of the Curve**:
- Identify intervals of concave up and concave down using the second derivative.
8. **Identify Asymptotes**:
- Investigate any vertical or horizontal asymptotes.
9. **Plot Key Points**:
- Locate important points such as intercepts (x and y), critical points, and inflection points.
10. **Find Absolute Extreme Points**:
- Determine and provide coordinates for any absolute maximum or minimum points, if they exist.
By following these steps, a comprehensive understanding and accurate graph of the function can be constructed.
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