THEOREM: Second Partials Test.

Let $\displaystyle f$ be a function of two variables with continuous second order partial derivatives in some disk centered at a critical point $\displaystyle (x_0,y_0)$, and let

$\displaystyle D=f_{xx}(x_0,y_0)f_{yy}(x_0,y_0)-f_{xy}^2$

(a) If $\displaystyle D>0$ and $\displaystyle f_{xx}(x_0,y_0)>0$, then $\displaystyle f$ has a relative minimum at $\displaystyle (x_0,y_0)$

(b) If $\displaystyle D>0$ and $\displaystyle f_{xx}(x_0,y_0)<0$, then $\displaystyle f$ has a relative maximum at $\displaystyle (x_0,y_0)$

(c) If $\displaystyle D<0$ , then $\displaystyle f$ has a saddle point at $\displaystyle (x_0,y_0)$

(d) If $\displaystyle D=0$, then no conclusion can be drawn