Partial Differentiation - Saddle Point II
Visualising gradient of z = f(x,y) in the x direction, the y direction, and the xy direction.
The equation for z = f(x,y) is:
The value of the coefficient is set by acoefficient in the applet. Use the slider to see the effect on the surface.
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).
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.
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.
Thus we get a saddlepoint.
The fact that there is a term combining x and y means that , and so we can use the Discriminant .
If D>0 then we have a local minimum (as the second derivative in the x direction, and the y direction, is positive).
If D<0, then the x*y term is of greater effect, so there is a saddle point.