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Given a polynomial f:R+→R+f: \mathbb{R}^+ \rightarrow \mathbb{R}^+f:R+→R+ that satisfies f(0)=1,f(1)=e,f(e)=πf(0) = 1, f(1) = e, f(e) =\pi f(0)=1,f(1)=e,f(e)=π.
Let I=∫1ef′(x)f(x) dx\displaystyle I = \int_1^e \frac{ f'(x) } { f(x) } \, dx I=∫1ef(x)f′(x)dx. Find the value of sgn(I)\text{sgn}(I) sgn(I).
Notation: sgn(x)\text{sgn}(x) sgn(x) denote the signum function, where sgn(x)={x∣x∣,x≠00,x=0 \text{sgn}(x) = \begin{cases} \dfrac{x}{|x|} , x \ne 0 \\ 0 , x = 0 \end{cases} sgn(x)=⎩⎨⎧∣x∣x,x=00,x=0.
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