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For f(z,y)=z²+²-42-10y +100 determine local extreme points if exists.

For f(z,y)=z²+²-42-10y +100 determine local extreme points if exists.

Calculus: Early Transcendentals
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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**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:**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). - \[\text{Options:}\] - \(\text{Check Answer/Save}\) - \(\text{Step-By-Step Example}\) - \(\text{Live Help}\)
### 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.
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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