Find the critical point(s) of the function. f(x, y) = x³y + 12x² − 8y + 4 - (Give your answer as a comma-separated list of points in the form (*,*) if needed. Express numbers notation and fractions where needed.) critical point(s):
Find the critical point(s) of the function. f(x, y) = x³y + 12x² − 8y + 4 - (Give your answer as a comma-separated list of points in the form (*,*) if needed. Express numbers notation and fractions where needed.) critical point(s):
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![**Problem Statement**
Find the critical point(s) of the function.
\[ f(x, y) = x^3 y + 12x^2 - 8y + 4 \]
(Give your answer as a comma-separated list of points in the form \((*,*)\) if needed. Express numbers in exact form. Use symbolic notation and fractions where needed.)
**Solution Format**
Critical point(s): \_\_\_\_\_\_\_\_
---
**Explanation**
In this problem, you are asked to find the critical points of the given function \( f(x, y) \). Critical points occur where the partial derivatives of the function with respect to \( x \) and \( y \) are both equal to zero.
To find the partial derivatives and solve for the points \((x_0, y_0)\), follow these steps:
1. Compute the partial derivative of \( f \) with respect to \( x \):
\[ f_x(x, y) = \frac{\partial}{\partial x} \left( x^3 y + 12x^2 - 8y + 4 \right) \]
2. Compute the partial derivative of \( f \) with respect to \( y \):
\[ f_y(x, y) = \frac{\partial}{\partial y} \left( x^3 y + 12x^2 - 8y + 4 \right) \]
3. Set the partial derivatives equal to zero and solve the resulting system of equations:
\[ f_x(x, y) = 0 \]
\[ f_y(x, y) = 0 \]
4. Find the values of \( x \) and \( y \) that satisfy both equations.
The result will be the critical points of the function.
Please type the critical points in the field provided using exact forms, such as fractions or integer values.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe132a23b-2bfe-4f64-9635-6f1845f8e4fa%2F6256b1aa-23d0-40a7-8064-3b3b05e73054%2Fzkuy9mq_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement**
Find the critical point(s) of the function.
\[ f(x, y) = x^3 y + 12x^2 - 8y + 4 \]
(Give your answer as a comma-separated list of points in the form \((*,*)\) if needed. Express numbers in exact form. Use symbolic notation and fractions where needed.)
**Solution Format**
Critical point(s): \_\_\_\_\_\_\_\_
---
**Explanation**
In this problem, you are asked to find the critical points of the given function \( f(x, y) \). Critical points occur where the partial derivatives of the function with respect to \( x \) and \( y \) are both equal to zero.
To find the partial derivatives and solve for the points \((x_0, y_0)\), follow these steps:
1. Compute the partial derivative of \( f \) with respect to \( x \):
\[ f_x(x, y) = \frac{\partial}{\partial x} \left( x^3 y + 12x^2 - 8y + 4 \right) \]
2. Compute the partial derivative of \( f \) with respect to \( y \):
\[ f_y(x, y) = \frac{\partial}{\partial y} \left( x^3 y + 12x^2 - 8y + 4 \right) \]
3. Set the partial derivatives equal to zero and solve the resulting system of equations:
\[ f_x(x, y) = 0 \]
\[ f_y(x, y) = 0 \]
4. Find the values of \( x \) and \( y \) that satisfy both equations.
The result will be the critical points of the function.
Please type the critical points in the field provided using exact forms, such as fractions or integer values.
Transcribed Image Text:### Categorize the Critical Point(s)
Select the appropriate category for the given critical point(s) from the options below:
- **local maximum**
- **local minimum**
- **saddle point**
- **none of the above**
This task helps to identify and classify critical points in functions, which are essential concepts in calculus and optimization.
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