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#1 2007-11-08 05:13:08

tony123
Member
Registered: 2007-08-03
Posts: 228

How far is the turning point K

The sum of the non-negative real numbers

is 2 and
. Find the largest and smallest possible values of

Last edited by tony123 (2007-11-08 06:27:29)

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#2 2007-11-08 10:42:56

JaneFairfax
Member
Registered: 2007-02-23
Posts: 6,868

Re: How far is the turning point K

Use the Cauchy–Schwarz inequality.

Last edited by JaneFairfax (2007-12-18 14:52:42)

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#3 2007-11-08 10:56:05

mikau
Member
Registered: 2005-08-22
Posts: 1,504

Re: How far is the turning point K

the Cauchy-Schwartz inequality - its everywhere!


A logarithm is just a misspelled algorithm.

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#4 2007-11-08 18:28:47

JaneFairfax
Member
Registered: 2007-02-23
Posts: 6,868

Re: How far is the turning point K

The Cauchy–Schwarz inequality is

Taking

we get

Taking

we get

Since both inequalities have to be satisfied, we take the first one: S ≥ 1. So the minimum value is 1.

Also

To show that this is attainable, take

. So the maximum value of S is 2.

Last edited by JaneFairfax (2007-11-08 18:29:03)

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#5 2007-11-08 20:18:20

JaneFairfax
Member
Registered: 2007-02-23
Posts: 6,868

Re: How far is the turning point K

Umm, sorry. The first part of my working is wrong. I’ve been trying to find an example for which S = 1, but I’ve come to realize that this is impossible. swear

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