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Question Number 183712 by Michaelfaraday last updated on 29/Dec/22

$$solve:$$$$W\left(In\left(4x\right)\right)=\sqrt{(x-1)}$$

Answered by mr W last updated on 29/Dec/22

$$e^{\ln \left(4x\right)}\ln \left(4x\right)=\sqrt{x-1}$$$${\left(4x\right)}^{4x}=e^{\sqrt{x-1}}$$$$lett=\sqrt{x-1}⩾0$$$${(4t^2+4)}^{(4t^2+4)}=e^t$$$$f\left(t\right)={(4t^2+4)}^{(4t^2+4)}$$$$g\left(t\right)=e^t$$$$f\left(0\right)-g\left(0\right)=4^4-e^0=256-1=255$$$$f^{\prime }\left(t\right)-g^{\prime }\left(t\right)=...>0$$$$\Rightarrow f\left(t\right)-g\left(t\right)isstrictlyincreasing,$$$$i.e.f\left(t\right)-g\left(t\right)⩾255fort⩾0$$$$\Rightarrow f\left(t\right)-g\left(t\right)=0hasnoroots!$$$$\Rightarrow W\left(In\left(4x\right)\right)=\sqrt{x-1}hasnoroots!$$

Commented by Michaelfaraday last updated on 29/Dec/22

$$wow,thankssirforyourhelp$$$$$$

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