Multivariable Second Derivative Test

Multivariable Second Derivative Test

In single variable calculus, we used the second derivative test as an alternative to doing the first derivative test to determine minima/maxima. Now, this is the multivariable version.

$$\text{Let } g(x,y) = f_{xx}(x,y)f_{yy}(x,y) - \left(f_{xy}(x,y)\right)^2$$ This occurs at a critical point, because we know this could be an extremum, we just want to know which kind. So $f_x = 0$ and $f_y = 0$

g > 0

$$g > 0 \text{ and } f_{xx} < 0:$$ $$\text{Local max}$$ $$g > 0 \text{ and } f_{xx} > 0:$$ $$\text{Local min}$$ In this case, $f_{xx}$ can't be 0, or else g wouldn't be > 0.

g < 0

It's a saddle point (think of a horse's saddle or a hyperbolic paraboloid).