The equation for z = f(x,y) is:[br][br][math]z=x^2+y^2-axy.[/math][br][br]The value of the coefficient is set by acoefficient in the applet. Use the slider to see the effect on the surface.[br][br]At the critical point, the curvature in the x direction is increasing (concave up), and the curvature in the y direction is increasing (concave up).[br][br]If the size of acoefficient is small, then the sum of the x^2 and y^2 terms are greater than the xy term, and so there will be a local minimum.[br][br]However, if the size of the acoefficient is large enough, then the xy term will be greater than the sum of x^2 and y^2. Then, in Quadrants 2 and 4 in the x-y plane, x*y is negative, so -a*x*y is positive, and the surface curves up. But, in Quadrants 1 and 3, x*y is positive, so -a*x*y is negative, and so the surface curves down.[br][br]Thus we get a saddlepoint.[br][br]The fact that there is a term combining x and y means that [math]\frac{\partial^2z}{\partial x\partial y}\ne0[/math], and so we can use the Discriminant [math]D=\left(\frac{\partial^2z}{\partial x^2}\right)\left(\frac{\partial^2z}{\partial y^2}\right)-\left(\frac{\partial^2z}{\partial x\partial y}\right)^2[/math].[br][br] If D>0 then we have a local minimum (as the second derivative in the x direction, and the y direction, is positive).[br][br]If D<0, then the x*y term is of greater effect, so there is a saddle point.[br][br]