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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 $\text{sgn}(x) = \begin{cases} \dfrac{x}{|x|} , x \ne 0 \\ 0 , x = 0 \end{cases}$ .