Second-derivative test and Hessian conditions

Forum for the GRE subject test in mathematics.
fullofquestions
Posts: 18
Joined: Sun Oct 07, 2007 1:49 pm

Second-derivative test and Hessian conditions

Postby fullofquestions » Sat Oct 13, 2007 4:51 pm

I am looking at the Max/Min problems section of the Princeton Review book and there is mention of the second derivative test involving calculation of the Hessian Matrix of a function f(x,y). Apparently, the following conclusions can be made:

if Hessian > 0 and d/dxdx (f) @ point P < 0 then f attains a max at P
if Hessian > 0 and d/dxdx (f) @ point P > 0 then f attains a min at P
if Hessian < 0, then f has a saddlepoint at P
if Hessian = 0, then no conclusion can be drawn

My only question is, when would I replace d/dxdx with d/dydy when drawing my conclusions? There is nothing about the problem that explains looking only at the second derivative with respect to x of f.

Just in case if you were wondering, I am using the Princeton Review 'Cracking the GRE Math Subject Test 3rd ed."

Thank you in advance.

heybob
Posts: 9
Joined: Mon Jul 23, 2007 12:41 am

Postby heybob » Sun Oct 14, 2007 8:00 pm

I do not believe you ever do. Just by the theory behind 3-D calculus, I believe you always check the derivative with respect to x. Don't quote me on that though....




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