Problem 4 - Calculus Early Transcendentals

Question

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.

Step-by-Step Solution

Answer

Answer: No, having both partial derivatives equal to zero does not necessarily imply a local maximum or local minimum. The second partial derivative test should be used to determine the nature of the critical point \((a, b)\), considering the determinant \(D(a, b)\) and the second partial derivatives \(f_{xx}(a, b)\) and \(f_{yy}(a, b)\).

Step 1: Statement of the second partial derivative test

The second partial derivative test states that if \(f_x(a, b) = f_y(a, b) = 0\), then we can consider the determinant \(D(a, b)\) defined as follows: $$ D(a, b) = \begin{vmatrix} f_{xx}(a, b) & f_{xy}(a, b) \\ f_{yx}(a, b) & f_{yy}(a, b) \end{vmatrix} = f_{xx}(a, b)f_{yy}(a, b) - (f_{xy}(a, b))^2 $$ Where \(f_{xx}, f_{yy}\) are the second partial derivatives of \(f\) with respect to 'x' and 'y', and \(f_{xy}, f_{yx}\) are the mixed partial derivatives. The outcome of the test can be summarized as follows: - 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 \(D(a, b) = 0\), the test is inconclusive.

Step 2: Explanation using the second partial derivative test

Given that both partial derivatives (\(f_x(a, b)\) and \(f_y(a, b)\)) are zero, the second partial derivative test indicates that this information alone is not sufficient to conclude whether \(f\) has a local maximum or local minimum at \((a, b)\). Instead, the values of the determinant \(D(a, b)\) and second partial derivatives (\(f_{xx}(a, b)\) and \(f_{yy}(a, b)\)) must also be considered. Hence, it is not guaranteed that \(f\) will have a local maximum or local minimum at \((a, b)\) when both first-order partial derivatives are zero. The second partial derivative test should be used to determine the nature of the critical point \((a, b)\).