Principles of Mathematical Analysis, Rudin, 3th ed, Chapter 5, Problem 4

Problem

If

$$ C_0 + \frac{C_1}{2} + \cdots + \frac{C_{n-1}}{n} + \frac{C_n}{n+1} =0, $$

where $C_0,\cdots,C_n$ are real constants, prove that the equation

$$ C_0 + C_1x + \cdots + C_{n-1}x^{n-1} + C_nx^n = 0 $$

has at least one real root between 0 and 1.

Answer

Define a function $p:[0,1] \rightarrow \mathbb{R}$ as

$$ p(x) = C_0x + \frac{C_1}{2}x^2 + \cdots + \frac{C_n}{n+1}x^{n+1}. $$

Then, $p$ is differentiable on $[0,1]$ and $p(0) = p(1) = 0$. By the mean value theorem, there exists $x \in (0,1)$ such that

$$ C_0 + C_1x + \cdots + C_{n-1}x^{n-1} + C_nx^n = p'(x) = \frac{p(1) - p(0)}{1-0} = 0.$$

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