Math Is Fun Forum

Suresh pallam

Guest

Help

If a,b,c,d are non-negative real numbers such that (a+b+c+d) =1, then
prove that (ab + bc + cd) cannot exceed 0.25

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Help

Hi;

I am sure this can be done with inequalities but it also can be done by maximizing (ab + bc + cd) using Lagrangian Multipliers.

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Srikantan

Member

From: Tanjore

Registered: 2016-06-04

Posts: 42

Re: Help

Can you pls explain in detail. I am not getting it. Thanks

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Help

Have you had some calculus and or inequalities?

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Srikantan

Member

From: Tanjore

Registered: 2016-06-04

Posts: 42

Re: Help

Yes i have but not sure how to use here

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Help

Have you seen the method of Lagrangian Multipliers?

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Srikantan

Member

From: Tanjore

Registered: 2016-06-04

Posts: 42

Re: Help

No bro sorry. Any logical lay man method there?

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Help

Could be, I will think about it while I get some fresh nighttime air. When I get back, if there is till no answer to this I will post my method and I will try to make it easy to understand. Of course, since I can do it, it is already easy.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Srikantan

Member

From: Tanjore

Registered: 2016-06-04

Posts: 42

Re: Help

Thanks a ton bro.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Help

By the method of Lagrangian Multipliers:

Form the system of equations that are made up of the partial derivatives of f equaling lambda times the partial derivative of g for a,b,c and d.

Performing the indicated differentiations and cleaning up a bit.

Adding g to that.

This system of equations as I once told the Legendary JaneFairfax will usually be very simple to solve and this set of equations is no different.

Solving we get:

we plug these values into f and get 1 / 4. Now there is a theorem that states that the value found by this method, in this case 1 / 4 is either a maximum or a minimum. In this case it is a maximum value for f. So, if the maximum value for f is 1 / 4, we have proven that f can never exceed .25

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

ElainaVW

Member

Registered: 2013-04-29

Posts: 580

Re: Help

Hello:

Where is the spiel?

Last edited by ElainaVW (2016-09-30 20:20:10)

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Help

Howdy;

Definition spiel - A lengthy or extravagant speech or argument usually intended to persuade. A rant.

You want the rant, then you shall have it, but not now.

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: Help

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{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

Srikantan

Member

From: Tanjore

Registered: 2016-06-04

Posts: 42

Re: Help

Awesome. Thanks

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: Help

But you have to take some care.Note that each term of the given expression ab+bc+cd should appear with -ve sign on R.H.S. of eq.2. that requires a,b,c and d to be alternately + and -.i.e (a-b+c-d)^2 If you had been asked for the other group bd+da+ac you had to take (a+b-c-d)^2

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{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}