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Question Number 165550 by mnjuly1970 last updated on 03/Feb/22

Answered by mahdipoor last updated on 03/Feb/22

$$f\left(1\right)=\sqrt{11},f\left(f\left(1\right)\right)=\sqrt{22},f\left(f\left(f\left(1\right)\right)\right)=\sqrt{33}$$$$...f(f...f\left(1\right)...)=\sqrt{10n+1}$$$$\Rightarrow f^{\prime }\left(x\right)=\frac{x}{\sqrt{x^2+10}}$$$$f^{\prime }\left(1\right)\times f\left(\sqrt{11}\right)\times f\left(\sqrt{22}\right)\times ...\times f\left(\sqrt{10n+1}\right)=$$$$\frac{1}{\sqrt{11}}\times \frac{\sqrt{11}}{\sqrt{22}}\times \frac{\sqrt{22}}{\sqrt{33}}\times ...\times \frac{\sqrt{10n+1}}{\sqrt{10n+11}}=$$$$\frac{1}{\sqrt{10n+11}}=\frac{1}{n+1}\Rightarrow n^2-8n-10=0\Rightarrow$$$$n=\frac{8\pm \sqrt{104}}{2}\notin \mathrm{N}\Rightarrow withuotanswer$$$$butif\frac{1}{\sqrt{10n+11}}=\frac{1}{n}\Rightarrow n^2-10n-11=0$$$$\Rightarrow n=11or-1\Rightarrow n=11$$

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