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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![### Finding Critical Points of a Multivariable Function
#### Problem Statement:
Find the local minima, local maxima, and saddle points for the following function:
\[ f(x, y) = 6x^2 - 2x^3 + 3y^2 + 6xy \]
#### Solution Outline:
To find the local minima, maxima, and saddle points, we follow these steps:
1. **Compute the First Derivatives:**
- Find \( \frac{\partial f}{\partial x} \) and \( \frac{\partial f}{\partial y} \).
2. **Set First Derivatives to Zero:**
- Solve the equations \( \frac{\partial f}{\partial x} = 0 \) and \( \frac{\partial f}{\partial y} = 0 \) to find critical points.
3. **Compute the Second Derivatives:**
- Find \( \frac{\partial^2 f}{\partial x^2} \), \( \frac{\partial^2 f}{\partial y^2} \), and \( \frac{\partial^2 f}{\partial x \partial y} \).
4. **Apply the Second Derivative Test:**
- Use the second derivatives to create the Hessian matrix.
- Calculate the determinant of the Hessian to determine the nature of each critical point:
- If \( D > 0 \) and \( \frac{\partial^2 f}{\partial x^2} > 0 \), it's a local minimum.
- If \( D > 0 \) and \( \frac{\partial^2 f}{\partial x^2} < 0 \), it's a local maximum.
- If \( D < 0 \), it's a saddle point.
- If \( D = 0 \), the test is inconclusive.
This systematic approach will identify all salient features of the function's graph in the neighborhood of its critical points.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd825c8c7-be90-4b8b-8df3-9fe3cc770042%2F9d29329a-1a20-483a-a7dc-8fc3e1299b7c%2Fybuwgx_processed.png&w=3840&q=75)
Transcribed Image Text:### Finding Critical Points of a Multivariable Function
#### Problem Statement:
Find the local minima, local maxima, and saddle points for the following function:
\[ f(x, y) = 6x^2 - 2x^3 + 3y^2 + 6xy \]
#### Solution Outline:
To find the local minima, maxima, and saddle points, we follow these steps:
1. **Compute the First Derivatives:**
- Find \( \frac{\partial f}{\partial x} \) and \( \frac{\partial f}{\partial y} \).
2. **Set First Derivatives to Zero:**
- Solve the equations \( \frac{\partial f}{\partial x} = 0 \) and \( \frac{\partial f}{\partial y} = 0 \) to find critical points.
3. **Compute the Second Derivatives:**
- Find \( \frac{\partial^2 f}{\partial x^2} \), \( \frac{\partial^2 f}{\partial y^2} \), and \( \frac{\partial^2 f}{\partial x \partial y} \).
4. **Apply the Second Derivative Test:**
- Use the second derivatives to create the Hessian matrix.
- Calculate the determinant of the Hessian to determine the nature of each critical point:
- If \( D > 0 \) and \( \frac{\partial^2 f}{\partial x^2} > 0 \), it's a local minimum.
- If \( D > 0 \) and \( \frac{\partial^2 f}{\partial x^2} < 0 \), it's a local maximum.
- If \( D < 0 \), it's a saddle point.
- If \( D = 0 \), the test is inconclusive.
This systematic approach will identify all salient features of the function's graph in the neighborhood of its critical points.
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