Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Let $c_0,c_1,c_2,\ldots ,c_n$ be constants such that :


I have to prove that the equation: $$c_0+c_1x+\ldots+c_nx^n=0$$

Has a real solution between 0 and 1.

Didn't know how to start...I thought that maybe I could use something about derivative...But I'm lost... Any help,much appreciated!!!

share|cite|improve this question
up vote 6 down vote accepted

HINT 1: $$c_0 + \dfrac{c_1}{2} + \cdots + \dfrac{c_n}{n+1} = \int_0^1 \left( c_0 + c_1x + \cdots + c_nx^n\right) dx$$


If $f(x)$ is continuous and $\displaystyle \int_a^b f(t) dt = 0$, then can you conclude that there is a root of $f(x)$ between $a$ and $b$?

share|cite|improve this answer
Thanks so much! – HipsterMathematician Sep 21 '12 at 15:19

This is an exercise from the differentiation chapter in Baby Rudin, and knowledge of integration is not expected. You can solve it without integrals using the following function:

$$ f(x) = c_0x+\frac{c_1x^2}{2}+\ldots+\frac{c_{n-1}x^n}{n}+\frac{c_nx^{n+1}}{n+1} $$

What value does it take at $0$ and $1$? What value must the derivative take in $(0, 1)$ then?

share|cite|improve this answer
Thanks a lot!!!Very helpful! – HipsterMathematician Sep 21 '12 at 15:16
This exercise also appears in Spivak's Calculus, if you should know. – Pedro Tamaroff Sep 29 '12 at 21:39
This is very much the same approach, you just wrote the integral. – Sawarnik Mar 5 '14 at 20:16

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.