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Chapter 12: Problem 4

If \(f_{x}(a, b)=f_{y}(a, b)=0,\) does it follow that \(f\) has a local maximum or local minimum at \((a, b) ?\) Explain.

Short Answer

Expert verified
Short Answer: No, having \(f_x(a, b) = f_y(a, b) = 0\) does not guarantee that the function \(f\) has a local maximum or local minimum at the point \((a, b)\). We need to perform the second partial derivative test and determine the type of critical point based on those results.

Step by step solution

01

Clarification of Notation

The given information can also be written as: - \(\frac{\partial f}{\partial x}(a, b) = 0\): The first partial derivative with respect to x at the point \((a, b)\) is zero. - \(\frac{\partial f}{\partial y}(a, b) = 0\): The first partial derivative with respect to y at the point \((a, b)\) is zero. #Step 2: The Second Partial Derivative Test#
02

The Second Partial Derivative Test

To determine if the function \(f\) has a local max, local min, or saddle at the point \((a, b)\), we need to evaluate the second partial derivatives at that point and use the second partial derivative test. Evaluate the following second partial derivatives at point \((a, b)\): - \(f_{xx}(a, b) = \frac{\partial^2 f}{\partial x^2}(a, b)\) - \(f_{yy}(a, b) = \frac{\partial^2 f}{\partial y^2}(a, b)\) - \(f_{xy}(a, b) = f_{yx}(a, b) = \frac{\partial^2 f}{\partial x\partial y}(a, b)\) #Step 3: Compute D(a, b)#
03

Compute D(a, b)

Compute D(a, b) using the following formula: \(D(a, b) = f_{xx}(a, b)f_{yy}(a, b) - f_{xy}(a, b)^2\) #Step 4: Determine the type of critical point#
04

Determine the type of critical point

Now, use D(a, b) to determine the type of critical point \((a, b)\): - If \(D(a, b) > 0\) and \(f_{xx}(a, b) > 0\), then \(f\) has a local minimum at \((a, b)\). - If \(D(a, b) > 0\) and \(f_{xx}(a, b) < 0\), then \(f\) has a local maximum at \((a, b)\). - If \(D(a, b) < 0\), then \(f\) has a saddle point at \((a, b)\). - If \(D(a, b) = 0\), then the second partial derivative test is inconclusive. Conclusion: From the steps above, we can conclude that just having \(f_x(a, b) = f_y(a, b) = 0\) is not enough to guarantee a local maximum or local minimum at \((a, b)\). We need to perform the second partial derivative test and determine the type of critical point based on those results.

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