Exercise 5.4

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Exercise 4: If $$C_0+\frac{C_1}{2}+\cdots+\frac{C_{n-1}}{n}+\frac{C_n}{n+1}=0,$$ where $C_0,\ldots,C_n$ are real constants, prove that the equation $$C_0+C_1x+\ldots+
    C_{n-1}x^{n-1}+C_nx^n=0$$ has at least one real root between 0 and 1.

ghostofgarborg · Accepted Answer

(	extit{Matt ``Frito'' Lundy}) \
    Let:
    $$f(x) = \sum_{i = 0}^n \frac{C_i x^{i+1}}{i+1}.$$
    Then we are looking for an $x \in (0,1)$ such that $f'(x) = 0$.
    But $f(x)$ is continuous in $[0,1]$, differentiable in $(0,1)$, $f(0) = 0$, and $f(1)= 0$.
    So by ``the'' mean value theorem, there exists an $x \in (0,1)$ such that $f'(x) = 0$, as desired.

Exercise 5.4

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Exercise 5.4

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