Math Is Fun Forum

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Root between (0,1)?

Prove that this polynomial has at least 1 root between 0 and 1.

Last edited by bobbym (2009-08-09 11:10:48)

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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.

TheDude

Member

Registered: 2007-10-23

Posts: 361

Re: Root between (0,1)?

Assume that there is a triplet (a, b, c) such that the polynomial has no roots between 0 and 1.  By the IVT we know that f(0) and f(1) must either both be greater than 0 or less than 0.  Since every term of the polynomial has a parameter we only need to consider the case where both are greater than 0.

Now solve for b:

Substitute these values into our previous inequality:

These inequality signs are strict, so we have a contradiction.  There is no triplet (a, b, c) where both f(0) and f(1) are greater than 0 (or less than 0), and so by the IVT there must be at least one root between 0 and 1 for all possible triplets.

Last edited by TheDude (2009-08-13 05:14:44)

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Kurre

Member

Registered: 2006-07-18

Posts: 280

Re: Root between (0,1)?

Im not following. You cant just substitute expressions for inequalities in other inequalities

\\

Substitute these values into our previous inequality:

The Right hand side is ok, but not the left side. You have:

if im not missing anything.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Root between (0,1)?

Hi TheDude;

There is also a much easier way to attack this problem then the IVT.

---

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.

TheDude

Member

Registered: 2007-10-23

Posts: 361

Re: Root between (0,1)?

Im not following. You cant just substitute expressions for inequalities in other inequalities

The Right hand side is ok, but not the left side. You have:

if im not missing anything.

Right?

Hi TheDude;

There is also a much easier way to attack this problem then the IVT.

Probably true, but I'm something of a 1-trick pony

Last edited by TheDude (2009-08-13 23:57:53)

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Wrap it in bacon

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Root between (0,1)?

Hi TheDude;

How about when b= -4 and c=1.

---

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.

Kurre

Member

Registered: 2006-07-18

Posts: 280

Re: Root between (0,1)?

Right?

No. You forget that there is a minus sign. We have 3c<-b. But:

there is a minus sign infront of the 3c/2, so we cant apply the inequality there.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Root between (0,1)?

Hi;

I thought that someone would get this. I even thought that I knew who it would be.

Here is the solution.

Now F(0) = F(1) = 0. Now by Rolle's theorem

must have some point d in (0,1) where f(d) =0. So d is a root in (0,1).

Last edited by bobbym (2009-08-24 01:20:52)

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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.