Landos Posted August 6, 2015 Share Posted August 6, 2015 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. Link to comment Share on other sites More sharing options...

Casey Wood Posted August 24, 2015 Share Posted August 24, 2015 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. Link to comment Share on other sites More sharing options...

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