Finding the Critical Point

Date: 05/09/2000 at 00:56:35
From: Veerawong Pipithsuksunt
Subject: Calculus - Critical point

Dear Dr. Math,

Find the critical point of

     f(x,y,z,t) = (x^2)*(y^2)*(z^2) + (t^2)*(x^2) + 3x

and determine whether the point is a maximum, minimum, or saddle 
point.

I can not find the critical point...

Thank you,
Veerawong

Date: 05/09/2000 at 08:23:47
From: Doctor Jerry
Subject: Re: Calculus - Critical point

Hi Veerawong,

I think that you cannot find a critical point because there is no 
critical point; that is, a point (x,y,z) for which each of the 
partials are 0.

(<> means not equal to)

t <> 0 is not possible because if this were true, then x = 0 
(because t*x^2 = 0); but this is impossible because 
3 + 2t^2*x + 2x*y^2*z^2 = 0.

t = 0 is not possible because if you check all the cases, each is 
impossible. For example, with t = 0, then consider x <> 0 and 
x = 0 cases; in the first, y = 0, but this cannot be because 
3 + 2x*y^2*z^2 = 0. And so on.

- Doctor Jerry, The Math Forum
  http://mathforum.org/dr.math/