From what I've gathered from my calculus supplements and the web, I want to know if I have the general computation procedure understood correctly.
Example: Given f such that f(x,y) = ___. Find and classify the critical points.
To begin, I find the first order partial derivatives of f. Which I will set to 0, and then use some factoring techniques. From this I will have critical point(s).
Then, I would find the second order partial derivatives of f, and place them accordingly into a 2x2 Hessian Matrix A (i.e. f_xx as a_11, f_xy as a_12, f_yx as a_21, fyy as a_22).
From here, I would sub in the critical point (granted there is only one C.P.) into the matrix A. Then, depending on how friendly the numbers of each entry turn out to be, I have one of two options to follow through to classify the points.
1.) If the matrix turns out to be a symmetric matrix, then I can just take the diagonal entries as corresponding eigenvalues, which can be used directly to classify critical points by the following...
Where all eigenvalues are positive is a positive definite which is local minimum.
Where all eigenvalues are negative is a negative definite which is local maximum.
Where it is not positive or negative definite, it is a saddle point.
2.) If the matrix is not a symmetric matrix, then I will need to verify that the each sub-matrices follows a certain condition to find it's classification such that
If det(a_1) > 0 and det(A) > 0, then it is a positive definite, which is a local minimum
If det(a_1) < 0 and det(A) < 0, then it is a negative definite, which is a local maximum
If det(a_1) > 0 and det(A) < 0 or the alternative, it is a saddle point.
Thus, the critical point is classified.
Does anybody know if this is the correct format for a general case of using Hessian to classify points? Thanks!