Find all points where the function has any relative extrema. Identify any saddle points. f(x,y) = 4x² + 6xy + 3y² - 4x-8 Find the derivatives fxx. fyy, and fxy. fxx =
Find all points where the function has any relative extrema. Identify any saddle points. f(x,y) = 4x² + 6xy + 3y² - 4x-8 Find the derivatives fxx. fyy, and fxy. fxx =
College Algebra
7th Edition
ISBN:9781305115545
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:James Stewart, Lothar Redlin, Saleem Watson
Chapter2: Functions
Section: Chapter Questions
Problem 20P: In these problem you are asked to find a function that models a real-life situation and then use the...
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![**Topic: Finding Relative Extrema and Saddle Points**
In this section, we will explore how to find all points where a given function has any relative extrema and identify any saddle points.
**Problem Statement:**
Given the function:
\[ f(x,y) = 4x^2 + 6xy + 3y^2 - 4x - 8 \]
**Task:**
1. Find all points where the function has any relative extrema.
2. Identify any saddle points.
**Steps to Solve:**
1. **Find the First Partial Derivatives:**
- \( f_x \) (partial derivative with respect to \( x \))
- \( f_y \) (partial derivative with respect to \( y \))
2. **Set the First Partial Derivatives to Zero:**
- Solve \( f_x = 0 \)
- Solve \( f_y = 0 \)
3. **Solve the System of Equations:**
- These will provide the critical points.
4. **Find the Second Partial Derivatives:**
- \( f_{xx} \) (second partial derivative with respect to \( x \))
- \( f_{yy} \) (second partial derivative with respect to \( y \))
- \( f_{xy} \) (mixed second partial derivative)
**Explicit Calculation Step:**
Next, calculate the second partial derivatives as asked.
**Second Partial Derivatives:**
\[ f_{xx} = \]
*Here, you would input the value of the second derivative with respect to \( x \).*
\[ f_{yy} = \]
*Here, you would input the value of the second derivative with respect to \( y \).*
\[ f_{xy} = \]
*Here, you would input the value of the mixed second derivative.*
**Determine the Type of Critical Points Using the Second Derivatives:**
Use the **second derivative test** for functions of two variables:
\[ D = f_{xx}(a,b)f_{yy}(a,b) - [f_{xy}(a,b)]^2 \]
- If \( D > 0 \) and \( f_{xx}(a,b) > 0 \), then \( f \) has a **local minimum** at \((a, b)\).
- If \( D > 0 \) and \( f_{xx}(a,b) < 0 \), then \( f \)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F756689fc-aa25-4045-a342-f553545a40f2%2Fd3e14c30-e60e-40ad-8c9d-fbec7be39b34%2Fqdljvup_processed.png&w=3840&q=75)
Transcribed Image Text:**Topic: Finding Relative Extrema and Saddle Points**
In this section, we will explore how to find all points where a given function has any relative extrema and identify any saddle points.
**Problem Statement:**
Given the function:
\[ f(x,y) = 4x^2 + 6xy + 3y^2 - 4x - 8 \]
**Task:**
1. Find all points where the function has any relative extrema.
2. Identify any saddle points.
**Steps to Solve:**
1. **Find the First Partial Derivatives:**
- \( f_x \) (partial derivative with respect to \( x \))
- \( f_y \) (partial derivative with respect to \( y \))
2. **Set the First Partial Derivatives to Zero:**
- Solve \( f_x = 0 \)
- Solve \( f_y = 0 \)
3. **Solve the System of Equations:**
- These will provide the critical points.
4. **Find the Second Partial Derivatives:**
- \( f_{xx} \) (second partial derivative with respect to \( x \))
- \( f_{yy} \) (second partial derivative with respect to \( y \))
- \( f_{xy} \) (mixed second partial derivative)
**Explicit Calculation Step:**
Next, calculate the second partial derivatives as asked.
**Second Partial Derivatives:**
\[ f_{xx} = \]
*Here, you would input the value of the second derivative with respect to \( x \).*
\[ f_{yy} = \]
*Here, you would input the value of the second derivative with respect to \( y \).*
\[ f_{xy} = \]
*Here, you would input the value of the mixed second derivative.*
**Determine the Type of Critical Points Using the Second Derivatives:**
Use the **second derivative test** for functions of two variables:
\[ D = f_{xx}(a,b)f_{yy}(a,b) - [f_{xy}(a,b)]^2 \]
- If \( D > 0 \) and \( f_{xx}(a,b) > 0 \), then \( f \) has a **local minimum** at \((a, b)\).
- If \( D > 0 \) and \( f_{xx}(a,b) < 0 \), then \( f \)
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