91Ó°ÊÓ

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.

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 \((a, b)\).