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Question Number 28124 by ajfour last updated on 20/Jan/18

$$f(R^{+}\to R)isadifferentiable$$$$functionobeying$$$$2f\left(x\right)=f\left(xy\right)+f\left(\frac{x}{y}\right)$$$$forallx,y\in R^{+}and$$$$f\left(1\right)=0,f{}^{\prime }\left(1\right)=1.$$$$Findf\left(x\right).Morequestionsmay$$$$follow..$$

Commented by prakash jain last updated on 20/Jan/18

$$y=x$$$$f\left(x^2\right)=2f\left(x\right)$$$$f\left(x\right)=c\ln x$$$$check$$$$2c\ln x=c\ln x+c\ln y+c\ln x-c\ln y$$$$f^{\prime }\left(x\right)=\frac{c}{x}$$$$f^{\prime }\left(1\right)=1\Rightarrow c=1$$$$f\left(x\right)=\ln x$$

Commented by ajfour last updated on 20/Jan/18

$$Arethefollowingtrue:$$$$\left(i\right)No.ofsolutionsof{f}^{-1}\left(x\right)=x^5$$$$is3$$$$\left(ii\right)thereisatleastonepoint\alpha \in$$$$(0,\mathrm{\infty })forwhich{f}^{-1}\left(x\right)-x^5>0$$$$forx\in (\alpha ,\mathrm{\infty })?$$

Commented by ajfour last updated on 20/Jan/18

$$thankyouSir.$$

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