Critical Point Test

Suppose $x_0,y_0$ is a critical point of $f(x,y)$ where both partial derivatives vanish. Further suppose that $f_x,f_y,f_{xx},f_{xy},f_{yy}$ are all continuous. Define:

$$A=f_{xx}(x_0,y_0),B=f_{xy}(x_0,y_0),C=f_{yy}(x_0,y_0)$$

Then, we have:

• If $B^2-AC<0$ and $A<0$ then $f(x,y)$ has a relative maximum at $(x_0,y_0)$
• If $B^2-AC<0$ and $A>0$ then $f(x,y)$ has a relative minimum at $(x_0,y_0)$
• If $B^2-AC>0$ then $f(x,y)$ has a saddle point at $(x_0,y_0)$
• If $B^2-AC=0$ then no conclusion can be made