Quadratic approximation
Quadratic approximation is used to determine the change in a function f of (X,y) when x and y are changed a little bit. It can be expressed as the sum of two terms: the first-order term and the linear term. The first-order term is f sub x times the X change in X, while the linear term is f sub y times the change in y. f the solution is at the critical point, then the derivative of the solution will be 0 at the critical point. So that term will go away, and that term will also be 0 at the critical point, so that term will also go away. The next term, a quadratic term, is derived from the Taylor expansion of a function of a single variable. The first derivative times x minus x0 plus 1/2 of the second derivative times x minus x0 squared equals the nth derivative times xn-1. And see, this side here is Taylor approximation in one variable looking only at x. But of course we also have terms involving y and terms involving simultaneously x and y. And these terms are f xy times change in x times change in y plus 1/2 fyy times y minus yO squared. There’s no 1/2 in the middle because, in fact, you would have two terms, one for xy, one for yx but they are the same. To make the approximation more accurate, one must consider cubic terms involving the third derivative and so on, but we are not actually looking at them. And sO now when we do this approximation well, the type of critical point remains the same when we replace the function with this approximation. And sO we can apply the argument that we used to deduce things in the quadratic case. In fact, it still works in general using this formula for approximation. We previously referred to the coefficient a as little a and called b capital B and c capital C. When you substitute these definitions into the various cases we had discussed, you end up with the second derivative test. In the degenerate case, we cannot say that a critical point is always a maximum or minimum. The approximation formula is justified only if higher-order terms are negligible. Quadratic approximation ΔF~~f_x*(x-x_0)+f_y*(y-y_0) +1/2f_xx(x-x_0)^2+f_xy(x-x_0)(y-y_0)+1/2f_yy(y-y_0)^2 -> – so the general case reduces to the quadratic case. In degenerate case, what actually happens depends on higher order derivatives In the case of non-degenerate functions, the shape of the graph is determined by the second and third derivatives, not by higher order derivatives. In the degenerate case, where the function is extremely flat or has a very small derivative in a certain region. the sign of the function’s second derivative can change from plus to minus or minus to plus. This changes the sign of the value of the function at that point, which can either make it a minimum (where it stops going down) or a saddle point (where it doesn’t go down at all). In real life, you have to be extremely lucky for this quantity to end up being exactly O. If it happens, then what you should do is maybe try by inspection, see if there’s a good reason the function should always be positive or always be negative or something or plot it on a computer and see what happens.
