Rolle's theorem for finding solutions

Question

Consider the equation: $ c_0 + c_1x + \cdots + c_{n-1} + c_{n} = 0$ where each $c_i \in R$

Given that: $ c_0 + \frac{c_1}{2} + \frac{c_3}{3} + \cdots + \frac{c_{n-1}}{n} + \frac{c_{n}}{n+1} = 0$

Prove that the equation has a root in the interval (0,1).

Hint: Use Rolle's theorem.

I'm stuck with this problem. I would usually attack this kind of problem's using Bolzano, but I don't know how to relate the derivative (as stated in Rolle's theorem) with the roots of the original equation.

Thanks in advance.

Did you mean $c_0 + c_1 x + \cdots + c_{n-1} x^{n-1}+ c_n x^n=0$? — Dustan Levenstein, Nov 17 '15 at 00:11

score 1 · Accepted Answer · answered Nov 17 '15 at 00:13

1

This is classic textbook problem.

Hint: Consider $f(x)=c_0x + \frac{c_1}{2}x^2+ \frac{c_3}{3}x^3 + \cdots + \frac{c_{n-1}}{n}x^n + \frac{c_{n}}{n+1}x^{n+1}$.

answered Nov 17 '15 at 00:13

lhf

216,483

Yes it follows this way! Thanks man! – Jordi Bosch Nov 17 '15 at 11:02

Rolle's theorem for finding solutions

1 Answers1