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Calculus Level 2

Given a polynomial \(f: \mathbb{R}^+ \rightarrow \mathbb{R}^+\) that satisfies \(f(0) = 1, f(1) = e, f(e) =\pi \).

Let \(\displaystyle I = \int_1^e \frac{ f'(x) } { f(x) } \, dx \). Find the value of \(\text{sgn}(I) \).

Notation: \(\text{sgn}(x) \) denote the signum function, where \( \text{sgn}(x) = \begin{cases} \dfrac{x}{|x|} , x \ne 0 \\ 0 , x = 0 \end{cases} \).

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