Question and Answers Forum

All Questions      Topic List

Algebra Questions

Previous in All Question      Next in All Question      

Previous in Algebra      Next in Algebra      

Question Number 180856 by Shrinava last updated on 18/Nov/22

$$\mathrm{find}\mathrm{all}\mathrm{values}\mathrm{of}\mathrm{m}\in \mathbb{R}\mathrm{such}\mathrm{that}\mathrm{the}\mathrm{equation}:$$$${\int }_0^{\mathbf{x}}\frac{\arctan \mathbf{y}}{\mathrm{y}}\mathrm{dy}=\mathrm{mx}$$$$\mathrm{has}\mathrm{two}\mathrm{real}\mathrm{roots}:{\mathrm{x}}_1\in (-\mathrm{\infty };0),{\mathrm{x}}_2\in (0;\mathrm{\infty })$$

Answered by mr W last updated on 19/Nov/22

$$f\left(x\right)={\int }_0^x\frac{{\tan }^{-1}t}{t}dt$$$$g\left(x\right)=mx$$$$wecanseef\left(x\right)isanoddfunction,$$$$i.e.f(-x)=-f\left(x\right).$$$$f^{\prime }\left(x\right)=\frac{{\tan }^{-1}x}{x}$$$$f^{\prime }\left(0\right)=\underset{x\to 0}{\lim }\frac{{\tan }^{-1}x}{x}=1$$$$i.e.thetangentofy=f\left(x\right)atx=0is$$$$y=kxwithk=1.$$$$\underset{x\to +\mathrm{\infty }}{\lim }{f}^{\prime }\left(x\right)=\underset{x\to +\mathrm{\infty }}{\lim }\frac{{\tan }^{-1}x}{x}=0$$$$sowecanknowhowthegraphof$$$$y=f\left(x\right)nearlylookslike.theline$$$$y=mxalwaysintersectsthecurve$$$$y=f\left(x\right)atx=0andadditionallyat$$$$twopointsPand{P}^{\prime }withx=\pm a$$$$ify=mxisflaterthanthetangent$$$$lineatx=0,i.e.if0<m<k=1.$$$$thatmeansif0<m<1,theeqn.$$$$f\left(x\right)=g\left(x\right)willhavetworoots:$$$$x=\pm a.sotheansweris0<m<1.$$

Commented by mr W last updated on 19/Nov/22

Commented by mr W last updated on 19/Nov/22

$$forthesolitionofthequestion,we$$$${don}^{\prime }tneedtosolvetheintegral$$$${\int }_0^x\frac{{\tan }^{-1}t}{t}dtreally.tobehonest,i{can}^{\prime }t$$$$solveit.but{it}^{\prime }senoughtoknowhow$$$$itsgraphnearlylookslike.$$

Commented by Shrinava last updated on 19/Nov/22

$$\mathrm{perfect}\mathrm{dear}\mathrm{professor}\mathrm{thank}\mathrm{you}$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com