Find and classify (local maximum, local mi points of f(x,y) = x2 - x²y + 2y². Show all supporting work to completely justify your %3D responses.
Find and classify (local maximum, local mi points of f(x,y) = x2 - x²y + 2y². Show all supporting work to completely justify your %3D responses.
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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![### Critical Points Classification: Analyzing f(x,y)
#### Problem Statement
Find and classify (local maximum, local minimum, saddle point) all critical points of the function:
\[ f(x,y) = x^2 - x^2y + 2y^2 \]
Show all supporting work to completely justify your responses.
#### Task Outline
1. **Find the Partial Derivatives**: Compute the first-order partial derivatives of \( f(x,y) \) with respect to \( x \) and \( y \).
2. **Solve for Critical Points**: Set these partial derivatives equal to zero and solve for \( x \) and \( y \).
3. **Classify the Critical Points**: Use second-order partial derivatives and the second derivative test to classify each critical point found.
Use detailed step-by-step calculations and ensure all solutions are justified logically and mathematically.
#### Detailed Solution Steps
1. **Compute \( f_x(x,y) = \frac{\partial f}{\partial x} \)**
\[
f_x(x,y) = \frac{\partial}{\partial x} (x^2 - x^2y + 2y^2)
\]
2. **Compute \( f_y(x,y) = \frac{\partial f}{\partial y} \)**
\[
f_y(x,y) = \frac{\partial}{\partial y} (x^2 - x^2y + 2y^2)
\]
3. **Find Critical Points**
\[
f_x(x,y) = 0
\]
\[
f_y(x,y) = 0
\]
4. **Second Partial Derivatives**
\[
f_{xx}(x,y) = \frac{\partial^2 f}{\partial x^2}
\]
\[
f_{yy}(x,y) = \frac{\partial^2 f}{\partial y^2}
\]
\[
f_{xy}(x,y) = \frac{\partial^2 f}{\partial x \partial y}
\]
5. **Second Derivative Test**:
\[
D = f_{xx}(a,b)f_{yy}(a,b) - [f_{xy}(a,b)]^2
\]
- If \( D > 0 \) and \(](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0bda9c1d-08ea-450e-a130-838d16131279%2Fd653e9bb-b1b6-4618-8fc7-ae8a66472e20%2Fig8mjfr_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Critical Points Classification: Analyzing f(x,y)
#### Problem Statement
Find and classify (local maximum, local minimum, saddle point) all critical points of the function:
\[ f(x,y) = x^2 - x^2y + 2y^2 \]
Show all supporting work to completely justify your responses.
#### Task Outline
1. **Find the Partial Derivatives**: Compute the first-order partial derivatives of \( f(x,y) \) with respect to \( x \) and \( y \).
2. **Solve for Critical Points**: Set these partial derivatives equal to zero and solve for \( x \) and \( y \).
3. **Classify the Critical Points**: Use second-order partial derivatives and the second derivative test to classify each critical point found.
Use detailed step-by-step calculations and ensure all solutions are justified logically and mathematically.
#### Detailed Solution Steps
1. **Compute \( f_x(x,y) = \frac{\partial f}{\partial x} \)**
\[
f_x(x,y) = \frac{\partial}{\partial x} (x^2 - x^2y + 2y^2)
\]
2. **Compute \( f_y(x,y) = \frac{\partial f}{\partial y} \)**
\[
f_y(x,y) = \frac{\partial}{\partial y} (x^2 - x^2y + 2y^2)
\]
3. **Find Critical Points**
\[
f_x(x,y) = 0
\]
\[
f_y(x,y) = 0
\]
4. **Second Partial Derivatives**
\[
f_{xx}(x,y) = \frac{\partial^2 f}{\partial x^2}
\]
\[
f_{yy}(x,y) = \frac{\partial^2 f}{\partial y^2}
\]
\[
f_{xy}(x,y) = \frac{\partial^2 f}{\partial x \partial y}
\]
5. **Second Derivative Test**:
\[
D = f_{xx}(a,b)f_{yy}(a,b) - [f_{xy}(a,b)]^2
\]
- If \( D > 0 \) and \(
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