Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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Question
1. Saddle-point at (4, 4) 2. Saddle-point at (4, 1) 3. Local maximum at (1, 1) 4. Local maximum at (4, 1) 5. Local minimum at (4, 1)
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Transcribed Image Text:1. Saddle-point at (4, 4) 2. Saddle-point at (4, 1) 3. Local maximum at (1, 1) 4. Local maximum at (4, 1) 5. Local minimum at (4, 1)
**Problem Statement:** Locate and classify the local extremum of the function \[ f(x, y) = x^2 + y^2 - 8x - 2y + 6. \] **Analysis:** To find the local extrema of the given function, we need to follow these steps: 1. **Find the Critical Points:** - Compute the partial derivatives of \( f(x, y) \) with respect to \( x \) and \( y \). - Set these partial derivatives to zero to find the critical points. 2. **Classify the Critical Points:** - Use the second derivative test for functions of two variables to classify each critical point. - Compute the second partial derivatives. - Calculate the determinant of the Hessian matrix. - Use the determinant to determine whether each critical point corresponds to a local minimum, local maximum, or saddle point. These are the steps involved in locating and classifying the local extrema of the given function of two variables.
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Transcribed Image Text:**Problem Statement:** Locate and classify the local extremum of the function \[ f(x, y) = x^2 + y^2 - 8x - 2y + 6. \] **Analysis:** To find the local extrema of the given function, we need to follow these steps: 1. **Find the Critical Points:** - Compute the partial derivatives of \( f(x, y) \) with respect to \( x \) and \( y \). - Set these partial derivatives to zero to find the critical points. 2. **Classify the Critical Points:** - Use the second derivative test for functions of two variables to classify each critical point. - Compute the second partial derivatives. - Calculate the determinant of the Hessian matrix. - Use the determinant to determine whether each critical point corresponds to a local minimum, local maximum, or saddle point. These are the steps involved in locating and classifying the local extrema of the given function of two variables.
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