Find all the critical points of the following functions and classify them as local maximum, local minimum or saddle point. (a) f(x,y) = x² + y² - 4x - 32y + 10 (b) f(x, y) = x³ + 6xy - 6x + y² - 2y.
Find all the critical points of the following functions and classify them as local maximum, local minimum or saddle point. (a) f(x,y) = x² + y² - 4x - 32y + 10 (b) f(x, y) = x³ + 6xy - 6x + y² - 2y.
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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![**Title: Finding Critical Points of Multivariable Functions**
In this section, we will learn how to find and classify the critical points of the following functions. Critical points are those points at which the first partial derivatives of a function vanish or are undefined. These points can be classified as local maxima, local minima, or saddle points.
**Problem Statement:**
Find all the critical points of the following functions and classify them as local maximum, local minimum, or saddle point.
**Functions:**
a) \( f(x, y) = x^4 + y^4 - 4x - 32y + 10 \)
b) \( f(x, y) = -e^{x^2} + 6xy - 6x + y^2 - 2y \)
**Solution Outline:**
1. **Find the partial derivatives** \( \frac{\partial f}{\partial x} \) and \( \frac{\partial f}{\partial y} \).
2. **Set the partial derivatives to zero** and solve the system of equations to find the critical points.
3. **Use the second partial derivative test** to classify each critical point found.
**Detailed Steps:**
1. **Partial Derivatives of \( f(x, y) \):**
- For \( f(x, y) = x^4 + y^4 - 4x - 32y + 10 \):
\[
\frac{\partial f}{\partial x} = 4x^3 - 4
\]
\[
\frac{\partial f}{\partial y} = 4y^3 - 32
\]
- For \( f(x, y) = -e^{x^2} + 6xy - 6x + y^2 - 2y \):
\[
\frac{\partial f}{\partial x} = -2xe^{x^2} + 6y - 6
\]
\[
\frac{\partial f}{\partial y} = 6x + 2y - 2
\]
2. **Set Partial Derivatives to Zero** and Solve:
- For \( f(x, y) = x^4 + y^4 - 4x - 32y + 10 \):
\[
4x^3 - 4](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F304238c1-892d-4f66-90f4-8d5405bfbe58%2Fcfa4cc45-7d8e-40e9-b08b-556f0fc7e5f4%2Fvtut77_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Finding Critical Points of Multivariable Functions**
In this section, we will learn how to find and classify the critical points of the following functions. Critical points are those points at which the first partial derivatives of a function vanish or are undefined. These points can be classified as local maxima, local minima, or saddle points.
**Problem Statement:**
Find all the critical points of the following functions and classify them as local maximum, local minimum, or saddle point.
**Functions:**
a) \( f(x, y) = x^4 + y^4 - 4x - 32y + 10 \)
b) \( f(x, y) = -e^{x^2} + 6xy - 6x + y^2 - 2y \)
**Solution Outline:**
1. **Find the partial derivatives** \( \frac{\partial f}{\partial x} \) and \( \frac{\partial f}{\partial y} \).
2. **Set the partial derivatives to zero** and solve the system of equations to find the critical points.
3. **Use the second partial derivative test** to classify each critical point found.
**Detailed Steps:**
1. **Partial Derivatives of \( f(x, y) \):**
- For \( f(x, y) = x^4 + y^4 - 4x - 32y + 10 \):
\[
\frac{\partial f}{\partial x} = 4x^3 - 4
\]
\[
\frac{\partial f}{\partial y} = 4y^3 - 32
\]
- For \( f(x, y) = -e^{x^2} + 6xy - 6x + y^2 - 2y \):
\[
\frac{\partial f}{\partial x} = -2xe^{x^2} + 6y - 6
\]
\[
\frac{\partial f}{\partial y} = 6x + 2y - 2
\]
2. **Set Partial Derivatives to Zero** and Solve:
- For \( f(x, y) = x^4 + y^4 - 4x - 32y + 10 \):
\[
4x^3 - 4
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