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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Question
![**Determining Local Extreme Points for the Function \(f(x, y)\)**
To find local extreme points for the function:
\[
f(x, y) = x^2 + y^2 - 4x - 10y + 100
\]
### Step 1: Determine the Conditions for Local Extreme Points
**Condition:**
If \((a, b)\) is a local extreme point for \(f(x, y)\), then it must be true that \( (f_x(a, b), f_y(a, b)) = (0,0) \).
**Input:**
0,0
- \[\text{Answer format: } a, b \text{ or } (a, b)\]
- **Status:** Correct (and saved).
- \[\text{Options:}\]
- \(\text{Check Answer/Save}\)
- \(\text{Step-By-Step Example}\)
- \(\text{Live Help}\)
### Step 2: Solve for \(z\)
**Condition:**
If we set \( f_z(x, y) = 0 \), then \( z \) is:
**Input:**
2
- \[\text{Answer format: Enter the value}\]
- **Status:** Correct (and saved).
- \[\text{Options:}\]
- \(\text{Check Answer/Save}\)
- \(\text{Step-By-Step Example}\)
- \(\text{Live Help}\)
### Step 3: Solve for \(y\)
**Condition:**
If we set \( f_y(x, y) = 0 \), then \( y \) is:
**Input:**
5
- \[\text{Answer format: Enter the value}\]
- **Status:** Correct (and saved).
- \[\text{Options:}\]
- \(\text{Check Answer/Save}\)
- \(\text{Step-By-Step Example}\)
- \(\text{Live Help}\)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc167378a-6772-4355-bdb0-a5fec1af8d0e%2Fe3a81b0e-9c5e-4069-aa69-77031d644614%2Fz5u0xdq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Determining Local Extreme Points for the Function \(f(x, y)\)**
To find local extreme points for the function:
\[
f(x, y) = x^2 + y^2 - 4x - 10y + 100
\]
### Step 1: Determine the Conditions for Local Extreme Points
**Condition:**
If \((a, b)\) is a local extreme point for \(f(x, y)\), then it must be true that \( (f_x(a, b), f_y(a, b)) = (0,0) \).
**Input:**
0,0
- \[\text{Answer format: } a, b \text{ or } (a, b)\]
- **Status:** Correct (and saved).
- \[\text{Options:}\]
- \(\text{Check Answer/Save}\)
- \(\text{Step-By-Step Example}\)
- \(\text{Live Help}\)
### Step 2: Solve for \(z\)
**Condition:**
If we set \( f_z(x, y) = 0 \), then \( z \) is:
**Input:**
2
- \[\text{Answer format: Enter the value}\]
- **Status:** Correct (and saved).
- \[\text{Options:}\]
- \(\text{Check Answer/Save}\)
- \(\text{Step-By-Step Example}\)
- \(\text{Live Help}\)
### Step 3: Solve for \(y\)
**Condition:**
If we set \( f_y(x, y) = 0 \), then \( y \) is:
**Input:**
5
- \[\text{Answer format: Enter the value}\]
- **Status:** Correct (and saved).
- \[\text{Options:}\]
- \(\text{Check Answer/Save}\)
- \(\text{Step-By-Step Example}\)
- \(\text{Live Help}\)
Transcribed Image Text:### Educational Website Content
#### Determining Values and Analyzing Points in Calculus
##### Problem Statement
**Question:**
What is the value of \((x, y)\) based on the values you got by setting \( f_x(x, y) = 0 \) and \( f_y(x, y) = 0 \)?
**Input:**
The answer was submitted as: \( (2, 5) \).
**Feedback:**
Answer format: Enter the value.
"Sorry, your answer is wrong (but saved). Better luck next time!"
**Options:**
- Check Answer/Save
- Step-By-Step Example
- Live Help
##### Analyzing Critical Points
**Question:**
The point you found is it a local max or min?
**Options:**
- ☐ It’s a local maximum
- ☑ It’s a local minimum
- ☐ It’s a saddle point; neither a max nor min
- ☐ It cannot be determined whether it is an extreme point
---
### Explanation
The problem involves using first partial derivatives, \( f_x(x, y) \) and \( f_y(x, y) \), to find critical points. Once the critical point is determined, you analyze whether it is a local maximum, a local minimum, or a saddle point. This exercise is crucial in optimization problems and understanding the behavior of multivariable functions.
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