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Question Number 197585 by Erico last updated on 23/Sep/23

$$\mathrm{f}\left(\mathrm{x}\right)=\mathrm{f}\left(\frac{1}{\mathrm{x}}\right)$$$$\Rightarrow \mathrm{f}\left(\mathrm{x}\right)=?$$

Answered by EdwarT last updated on 24/Sep/23

$$-\frac{1}{x^2}$$

Answered by Erico last updated on 24/Sep/23

$$\mathrm{May}\mathrm{be}\mathrm{like}\mathrm{this}$$$$\mathrm{f}\left(\mathrm{x}\right)=2\sqrt{\mathrm{x}}\cos (\frac{\sqrt{3}}{2}\mathrm{lnx}-\frac{\pi }{3})$$

Answered by Frix last updated on 24/Sep/23

$$\mathrm{For}x>0$$$$f\left(x\right)=2\sqrt{x}\sin \frac{2\pi +3\sqrt{3}\ln x}{6}$$

Answered by witcher3 last updated on 24/Sep/23

$${\mathrm{f}}^{\prime }\left(\mathrm{x}\right)=\mathrm{f}\left(\frac{1}{\mathrm{x}}\right)$$$${\mathrm{f}}^{″}\left(\mathrm{x}\right)=-\frac{1}{{\mathrm{x}}^2}{\mathrm{f}}^{\prime }\left(\frac{1}{\mathrm{x}}\right)=-\frac{1}{{\mathrm{x}}^2}\mathrm{f}\left(\mathrm{x}\right)$$$$\iff {\mathrm{x}}^2{\mathrm{f}}^{″}\left(\mathrm{x}\right)+\mathrm{f}\left(\mathrm{x}\right)=0$$$$\mathrm{f}\left(\mathrm{x}\right)={\mathrm{x}}^{\mathrm{r}}$$$$\Rightarrow (\mathrm{r}(\mathrm{r}-1)+1){\mathrm{x}}^{\mathrm{r}}=0$$$$\Rightarrow {\mathrm{r}}^2-\mathrm{r}+1=0$$$$\mathrm{r}=\frac{1+\mathrm{i}\sqrt{3}}{2},\frac{1-\mathrm{i}\sqrt{3}}{2}$$$$\mathrm{f}\left(\mathrm{x}\right)={\mathrm{x}}^{\frac{1}{2}}({\mathrm{ax}}^{\frac{\mathrm{i}}{2\sqrt{3}}}+{\mathrm{bx}}^{-\frac{\mathrm{i}}{2\sqrt{3}}})$$$$=\sqrt{\mathrm{x}}(\mathrm{asin}\left(\frac{\ln \left(\mathrm{x}\right)}{2}\sqrt{3}\right)+\mathrm{bcos}\left(\frac{\ln \left(\mathrm{x}\right)}{2}\sqrt{3}\right)$$$$$$$${\mathrm{f}}^{\prime }\left(\mathrm{x}\right)=\mathrm{f}\left(\frac{1}{\mathrm{x}}\right)$$$$\Rightarrow$$$$\frac{1}{\sqrt{\mathrm{x}}}\left((\frac{\mathrm{a}}{2}-\frac{\mathrm{b}}{2}\sqrt{3})\sin \left(\frac{\ln \left(\mathrm{x}\right)}{2}\right]\sqrt{3}\right)+(\frac{\mathrm{b}}{2}+\frac{\mathrm{a}}{2}\sqrt{3})\cos \left(\frac{\ln \left(\mathrm{x}\right)}{2}\sqrt{3}\right))$$$$=\frac{1}{\sqrt{\mathrm{x}}}(\mathrm{asin}\left(\frac{\ln \left(\frac{1}{\mathrm{x}}\right)}{2}\sqrt{3}\right)+\mathrm{bcos}\left(\frac{\ln \left(\frac{1}{\mathrm{x}}\right)}{2}\sqrt{3}\right)),\mathrm{\forall }\mathrm{x}>0$$$$\Rightarrow \frac{\mathrm{a}}{2}-\frac{\mathrm{b}}{2}\sqrt{3}=-\mathrm{a}$$$$\frac{\mathrm{b}}{2}+\frac{\mathrm{a}}{2}\sqrt{3}=\mathrm{b}$$$$\mathrm{b}=\mathrm{a}\sqrt{3}$$$$\mathrm{f}\left(\mathrm{x}\right)=\mathrm{a}\sqrt{\mathrm{x}}(\sin \left(\frac{\ln \left(\mathrm{x}\right)\sqrt{3}}{2}\right)+\sqrt{3}\cos \left(\frac{\ln \left(\mathrm{x}\right)\sqrt{3}}{2}\right))$$$$=2\mathrm{a}\sqrt{\mathrm{x}}\left(\sin (\frac{\ln \left(\mathrm{x}\right)\sqrt{3}}{2}+\frac{\pi }{3})\right),\mathrm{a}\in \mathbb{R}$$$$$$$$$$

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