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Topic: Try and find a geometrical proof for this one, George. (Or, anyone else!)
Replies: 10   Last Post: Mar 19, 2014 9:09 PM

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Port563

Posts: 668
Registered: 12/15/13
Try and find a geometrical proof for this one, George. (Or, anyone else!)
Posted: Mar 17, 2014 9:07 PM
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At MID <532668f0$0$504$815e3792@news.qwest.net>, George Cornelius very
quickly and elegantly found, via wholly geometric means, the minimum value
of (x+y)(x+z) where xyz(x+y+z)=1 and x,y,z>0 [all negative would also work].


So, try to find a geometric solution to this one, then... I cannot.


x, y, z > 0
1/3 <= xy + yz + zx <= 3
What are the highest and lowest possible values of each of:
(a) xyz
(b) x + y + z ?


Failing a geometric solution (which I frankly doubt exists, except perhaps
in a very artificial / contrived manner), let's see if an elegant "other"
method can be found.

Again, calculus is prohibited.

Intuition may get one to the answer quickly, but a rigorous proof is needed.





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