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## Homework Statement

Suppose a

_{n}/n+1 +...+a

_{0}/1=0.

Prove f(x) =a

_{n}x

^{n}+...+a

_{0}has a root between zero and one.

## Homework Equations

I'm pretty sure this is induction, but I'm not completely sure.

Mean Value Theorem probably

## The Attempt at a Solution

Well f(0)=a

_{0}and f(1)=a

_{n}+ ... + a

_{0}.

If it is induction, it is easy to show that when n=0, f(x)=0 for all x and thus it is obviously constant and has infinitely many roots between 0 and 1.

Assuming a root between 0 and 1 for n=k, the case for n=k+1 just doesn't seem very manipulable to me to create any kind of cancellation or truth arising from n=k. I'm pretty sure I have to manipulate the a

_{n}/n+1 +...+a

_{0}=0 deal, but how to do that I have no clue.

Also, should I aim to prove that f(0) and f(1) have opposite signs? This gives me the end result immediately from the Mean Value Theorem. How would I accomplish that? What if they are both zero?

Is induction the right method here, or am I barking up the wrong tree?

Any help would be much appreciated.