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Question Number 129319 by snipers237 last updated on 14/Jan/21
Prove that    if f is such as f ′(x)=((f(x))/(x(1−x−f(x))))  and f(1)=0 but f ≇Θ . Then   ★ f is the unique bijection from R^∗  to R and    ★lim_(x→0)  f(x)=+∞  and lim_(x→0) xf(x)=0   ★ ∫_0 ^(+∞) f^(−1) (y)dy= ζ(2)=∫_0 ^1 f(x)dx
$${Prove}\:{that}\: \\ $$$$\:{if}\:{f}\:{is}\:{such}\:{as}\:{f}\:'\left({x}\right)=\frac{{f}\left({x}\right)}{{x}\left(\mathrm{1}−{x}−{f}\left({x}\right)\right)} \\ $$$${and}\:{f}\left(\mathrm{1}\right)=\mathrm{0}\:{but}\:{f}\:\ncong\Theta\:.\:{Then} \\ $$$$\:\bigstar\:{f}\:{is}\:{the}\:{unique}\:{bijection}\:{from}\:\mathbb{R}^{\ast} \:{to}\:\mathbb{R}\:{and}\: \\ $$$$\:\bigstar\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:{f}\left({x}\right)=+\infty\:\:{and}\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}{xf}\left({x}\right)=\mathrm{0} \\ $$$$\:\bigstar\:\int_{\mathrm{0}} ^{+\infty} {f}^{−\mathrm{1}} \left({y}\right){dy}=\:\zeta\left(\mathrm{2}\right)=\int_{\mathrm{0}} ^{\mathrm{1}} {f}\left({x}\right){dx} \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$

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