Second derivative Test

We need to be able to determine whether or not a function has an extreme value at a critical point. The following test, which is proved at the end of this section, is analogous to the Second Derivative Test for functions of one variable.

Second Derivatives Test Suppose the second partial derivatives of \(f\) are continuous on a disk with center \((a, b)\), and suppose that \(f_x(a, b) = 0\) and \(f_y(a, b) = 0\) [that is, \((a, b)\) is a critical point of \(f\)]. Let \[ D = D(a, b) = f_{xx}(a, b)f_{yy}(a, b) - [f_{xy}(a, b)]^2 \] (a) If \(D > 0\) and \(f_{xx}(a, b) > 0\), then \(f(a, b)\) is a local minimum. (b) If \(D > 0\) and \(f_{xx}(a, b) < 0\), then \(f(a, b)\) is a local maximum. (c) If \(D < 0\), then \(f(a, b)\) is not a local maximum or minimum.

NOTE 1 In case (c) the point \((a, b)\) is called a saddle point of \(f\) and the graph of \(f\) crosses its tangent plane at \((a, b)\). NOTE 2 If \(D = 0\), the test gives no information: \(f\) could have a local maximum or local minimum at \((a, b)\), or \((a, b)\) could be a saddle point of \(f\). NOTE 3 To remember the formula for \(D\), it’s helpful to write it as a determinant: \[ D = \begin{vmatrix} f_{xx} & f_{xy} \\ f_{yx} & f_{yy} \end{vmatrix} = f_{xx}f_{yy} - (f_{xy})^2 \]

EXAMPLE Find the local maximum and minimum values and saddle points of \(f(x, y) = x^4 + y^4 - 4xy + 1\).

EXAMPLE Find and classify the critical points of the function \[ f(x, y) = 10x^2y - 5x^2 - 4y^2 - x^4 - 2y^4 \] Also find the highest point on the graph of \(f\).