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Question Number 85653 by M±th+et£s last updated on 23/Mar/20

$$iff⩾0and\frac{d}{dx}{\left(f\left(x\right)\right)}^2={\left(f^{\prime }\left(x\right)\right)}^{2}andf\left(0\right)=1$$$$findf\left(x\right)$$$$$$$$iff\left(x\right)=\frac{4x^3}{x^2+1}find{f}^{-1}\left(x\right)and{\left(f^{-1}\right)}^{{}^{\prime }}\left(2\right)$$

Commented by john santu last updated on 24/Mar/20

$$\left(1\right)2f\left(x\right)f{}^{\prime }\left(x\right)={\left(f{}^{\prime }\left(x\right)\right)}^2$$$$f{}^{\prime }\left(x\right)[f{}^{\prime }\left(x\right)-2f\left(x\right)]=0$$$$\Rightarrow f{}^{\prime }\left(x\right)=0\Rightarrow f\left(x\right)=k,k=constan$$$$f\left(0\right)=1\Rightarrow k=1\Rightarrow f\left(x\right)=1$$$$\Rightarrow f{}^{\prime }\left(x\right)=2f\left(x\right)$$$$df\left(x\right)=2f\left(x\right)dx$$$$\int df\left(x\right)=\int 2f\left(x\right)dx$$$$f\left(x\right)=2\int f\left(x\right)dx$$

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