Vector Calculus

Multivariate Extrema

Second Derivative 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 $f_y(a,b)=0$ i.e. $(a,b)$ is a critical point of $f$,

$$D=D(a,b)=f_{xx}(a,b)f_{yy}(a,b)-[f_{xy}(a,b){]}^2$$

• If $D>0$ and $f_{xx}(a,b)>0$: $f(a,b)$ is a local minimum.
• If $D>0$ and $f_{xx}(a,b)<0$: $f(a,b)$ is a local maximum.
• If $D<0$: $f(a,b)$ is a saddle point i.e. neither a local maximum or minimum.
• If $D=0$ then the test is inconclusive.

Use Determinant to Remember

$D=\left|\begin{array}{cc}{f}_{xx}& f_{xy}\\ f_{yx}& f_{yy}\end{array}\right|=f_{xx}{f}_{yy}-(f_{xy}{)}^2$

Critical Points

Points where all the derivatives are equal to zero or any of the derivatives are undefined.

Steps to Find

1. Find $\frac{\partial f}{\partial x},\frac{\partial f}{\partial y},...$
2. Set equal to zero