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Let f(r,y) r+ 12ry+12y. Find all the critical values and claesify them.

Let f(r,y) r+ 12ry+12y. Find all the critical values and claesify them.

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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**Problem 3: Critical Values of a Function** Let \( f(x, y) = x^2 + 12xy + 12y^3 \). Find all the critical values and classify them. **Solution Steps:** To find the critical values, we need to determine where the gradient (the vector of first partial derivatives) is equal to zero or is undefined. 1. **Partial Derivatives:** - Find \( \frac{\partial f}{\partial x} \) - Find \( \frac{\partial f}{\partial y} \) 2. **Set Partial Derivatives to Zero:** - Solve \( \frac{\partial f}{\partial x} = 0 \) - Solve \( \frac{\partial f}{\partial y} = 0 \) 3. **Critical Points:** - Identify the points \((x, y)\) where both partial derivatives are zero. 4. **Classify Critical Points:** - Use the second derivative test or Hessian matrix to determine whether each critical point is a local maximum, local minimum, or saddle point. This approach provides a foundation for analyzing the function and understanding the behavior of \( f(x, y) \) at its critical points.
Transcribed Image Text:**Problem 3: Critical Values of a Function** Let \( f(x, y) = x^2 + 12xy + 12y^3 \). Find all the critical values and classify them. **Solution Steps:** To find the critical values, we need to determine where the gradient (the vector of first partial derivatives) is equal to zero or is undefined. 1. **Partial Derivatives:** - Find \( \frac{\partial f}{\partial x} \) - Find \( \frac{\partial f}{\partial y} \) 2. **Set Partial Derivatives to Zero:** - Solve \( \frac{\partial f}{\partial x} = 0 \) - Solve \( \frac{\partial f}{\partial y} = 0 \) 3. **Critical Points:** - Identify the points \((x, y)\) where both partial derivatives are zero. 4. **Classify Critical Points:** - Use the second derivative test or Hessian matrix to determine whether each critical point is a local maximum, local minimum, or saddle point. This approach provides a foundation for analyzing the function and understanding the behavior of \( f(x, y) \) at its critical points.
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