17 Second derivative test I
17.1 Exploratory unit
- \(f_{xx}(x_0,y_0)\) measures the concavity (in the \(x\)-direction) of the function \(f\) at the point \((x_0,y_0)\).
- \(f_{yy}(x_0,y_0)\) measures the concavity (in the \(y\)-direction) of the function \(f\) at the point \((x_0,y_0)\).
- \(f_{xy}(x_0,y_0)\) measures the cross-concavity of the function \(f\) at the point \((x_0,y_0)\).
17.2 Main result
Suppose that \((x_0,y_0)\) is a critical point of \(f(x,y)\), and set \[ D = D(x_0, y_0) = f_{xx}(x_0,y_0)f_{yy}(x_0,y_0) - f_{xy}(x_0,y_0)^2\]
- If \(D > 0\), then \(f\) has a local extrema at \((x_0,y_0)\):
- If \(f_{xx}(x_0,y_0) > 0\), then \(f\) has a local minimum at \((x_0,y_0)\).
- If \(f_{xx}(x_0,y_0) < 0\), then \(f\) has a local maximum at \((x_0,y_0)\).
- If \(D < 0\), then \(f\) has a saddle point at \((x_0,y_0)\).
- If \(D = 0\), then the test is inconclusive.
If \(D\) is positive, i.e., if \(f\) has an extrema at \((x_0,y_0)\), then we must have
- \(f_{xx}(x_0,y_0) > 0\) and \(f_{yy}(x_0,y_0) > 0\), or
- \(f_{xx}(x_0,y_0) < 0\) and \(f_{yy}(x_0,y_0) < 0\).
This follows as the \(-f_{xy}^2\) term in \(D\) is always nonpositive, so \(f_{xx} f_{yy}\) must be positive to make \(D\) positive.
The \(D\) of the second derivative test is the determinant of a matrix (the Hessian): \[D = \begin{vmatrix} f_{xx} & f_{xy}\\ f_{yx} & f_{yy} \end{vmatrix} = f_{xx}f_{yy} - f_{xy}^2,\] where we lean on Clairaut’s theorem (\(f_{xy} = f_{yx}\)). As will be discussed in future courses, this interpretation of \(D\) provides geometric data in the language of linear algebra. While we won’t use this fact in this class, it is a handy way to remember the formula for \(D\)!