Intuition behind one second partial derivative test case

[Landos]

I want the geometrical interpretation of the following:

> If fxx * fyy} < fxy^2 and fxx has the same sign as fyy at a point, then why is that point a saddle point? Because,in the case that they have the same sign,one would expect that point to be a minimum or maximum point,not a saddle point.

Bear in mind,that i have checked the intuitive explanation that is available in wikipedia,but did not understand it.

Also, i am studying physics and not math,so please don't complicate things with symbolisms that i might not understand.

Also, i want an intuitive answer,because i know the maths behind categorizing critical points.

[Casey Wood]

hmmm... Is this an elliptic equation in the form ax^2 + 2bxy +cy^2 = 0, where a, b, and c are constants? If so then what you have is called a real quadratic curve. Such a curve makes either an ellipse, a hyperbola, or a parabola. It all depends on the sign of the quantity (ac-b^2). If (ac - b^2) is positive, then you have an ellipse, if it is negative you have a hyperbola, and if it is zero you have a parabola. Remember that elliptic equations have smooth solutions, and their stationary points are saddles, not maxima or minima.

 

Also remember the rules of the second partial derivative test, or D-Test:

 

Suppose that a function z = f ( x, y ) and the first partial derivatives and second partial derivatives are all defined in an open region R and that ( a, b ) is a stationary point in R such that

 

f_x ( a, b) = 0 and f_y ( a, b) = 0

 

Define the Quantity D as follows:

 

D = f_xx ( a, b) * f_yy ( a, b ) - [ f_xy ( a, b) ] ^ 2

 

Condition 1: If D > 0 and f_xx ( a, b ) > 0, then f ( a, b ) is a local minimum.

 

Condition 2: If D > 0 and f_xx ( a, b ) < 0, then f ( a, b ) is a local maximum.

 

Condition 3: If D < 0, then f ( a, b, f ( a, b ) ) is a saddle point.

 

Condition 4: If D = 0, then this test gives no information.

 

I highly recommend doing some curve sketching under these conditions to get a proper feel for the geometry of what is happening. Hope that helps ya out a bit.