NetNews Usenet Archive 1992 #16/NN_1992_16.iso/spool/sci/math/9614

Text File | 1992-07-28 | 1.7 KB | 50 lines

Newsgroups: sci.math
Path: sparky!uunet!elroy.jpl.nasa.gov!sdd.hp.com!cs.utexas.edu!asuvax!ukma!eng.ufl.edu!news
From: [e-mail] (Armando Barreto)
Subject: HELP. TEST FOR EXTREMA IN f(x,y,z)
Message-ID: <[e-mail]>
Keywords: Partial derivatives, Extrema, function of 3 variables.
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Date: Wed, 29 Jul 92 01:05:42 GMT
Lines: 38


Hello,

Most Calculus books include a theorem like this to test extrema of a function
of two variables (This one taken from Calculus ..  by Swokowsky):

Let f(x,y) be a function of two variables which has continuous second partial
derivatives on a rectangular region Q, and let

        g(x,y) = fxx(x,y)fyy(x,y) - [fxy(x,y)]^2

for all (x,y) in Q. If (a,b) is in Q and fx(a,b)=0, fy(a,b) =0, then the
following statements hold.

(i) f(a,b) is a local maximum if g(a,b)>0 and fxx(a,b) < 0.
(ii) f(a,b) is a local maximum if g(a,b)>0 and fxx(a,b) > 0.
(iii) f(a,b) is not an extremum of f if g(a,b)<0.

<<< NOTATION: fx(x,y) is the first partioal w.r.t. x, fxx(x,y) is the second
partial w.r.t. x and fxy(x,y) is the second mixed partial >>>

My question is :

IS THERE SUCH A TEST FOR EXTREMA IN A FUNCTION OF 3 VARIABLES :f(x,y,z) ?

I will appreciate any answers to the question or bibliographical references
that can guide me in finding the test myself.

Also, I tend to interpret the first term of g(x,y) as the amount of concavity
or convexity of f(x,y) at the test point (simultaneous curvature in x and y),
but, is there a :"graphical interpretration" for the second term of g(x,y) ?

Thanks in advance.

Armando Barreto     [e-mail]