Math Is Fun Forum

stevengoh911

Member

Registered: 2007-07-10

Posts: 1

Please solve my question

Given
x,y ∈ [0,2c]
f(x),g(y) ∈ [0,2c]
Show
| xy-f(x)+g(y)|≥ c²
exist

I don't really get this question.
Please help me

krassi_holmz

Real Member

Registered: 2005-12-02

Posts: 1,905

Re: Please solve my question

I think the question is:
Given two real numbers

and two functions

then show there is a numbers

such that

I think you should give more restrictions for f,g, because now it's only a mapping of [0,2c] to [0,2c]...
But I can't figure out an example right now.
I'll post later.
EDITED

Last edited by krassi_holmz (2007-07-10 09:17:45)

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IPBLE:  Increasing Performance By Lowering Expectations.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Please solve my question

Did you mean:

If that's the case, then let f(x) = 0, g(y) = 0, and x_0 = y_0 = c.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

krassi_holmz

Real Member

Registered: 2005-12-02

Posts: 1,905

Re: Please solve my question

Hi ricky.
No I mean

and you can't chose the functions - should prove it for all functions f, g : [0,2c]->[0,2c]

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IPBLE:  Increasing Performance By Lowering Expectations.

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Please solve my question

Note that for all x and y

.

If

, then

; then taking x[sub]0[/sub] = y[sub]0[/sub] = 2c will satisfy the inequality.

Also, if for all x

, then for all x and y

, in which case taking x[sub]0[/sub] = y[sub]0[/sub] = 0 will also ensure that

.

So it remains to consider the case

, i.e.

.

Sorry, I’m stuck here.

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Please solve my question

Duhhh, wait. Since for all x

, if the second scenario to holds, we would have

, which would reduce to the trivial case c = 0.

So we only need to consider the case

.

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Please solve my question

Anyone got any ideas yet?