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 173418 by Shrinava last updated on 11/Jul/22

Answered by aleks041103 last updated on 12/Jul/22

1) obvious soln. x=0  Observe that:  (t^2 /((t sinh(t) − cosh(t))^2 ))≥0 (equality for t≠0)  2) x>0  ∫_0 ^( x>0) f(t)dt>0, since f(t)>0 for t>0.  ⇒∄x>0, s.t. ∫_0 ^( x) ((t^2  dt)/((t sinh(t) − cosh(t))^2 ))=0  3) x<0  ∫_0 ^( x<0) f(t)dt=−∫_(x<0) ^( 0) f(t)dt<0  since f(t)>0 for t<0(t≠0)  ⇒∄x<0, s.t. ∫_0 ^( x) ((t^2  dt)/((t sinh(t) − cosh(t))^2 ))=0      ⇒Only soln. is x=0

$$\left.\mathrm{1}\right)\:{obvious}\:{soln}.\:{x}=\mathrm{0} \\ $$$${Observe}\:{that}: \\ $$$$\frac{{t}^{\mathrm{2}} }{\left({t}\:{sinh}\left({t}\right)\:−\:{cosh}\left({t}\right)\right)^{\mathrm{2}} }\geqslant\mathrm{0}\:\left({equality}\:{for}\:{t}\neq\mathrm{0}\right) \\ $$$$\left.\mathrm{2}\right)\:{x}>\mathrm{0} \\ $$$$\int_{\mathrm{0}} ^{\:{x}>\mathrm{0}} {f}\left({t}\right){dt}>\mathrm{0},\:{since}\:{f}\left({t}\right)>\mathrm{0}\:{for}\:{t}>\mathrm{0}. \\ $$$$\Rightarrow\nexists{x}>\mathrm{0},\:{s}.{t}.\:\int_{\mathrm{0}} ^{\:{x}} \frac{{t}^{\mathrm{2}} \:{dt}}{\left({t}\:{sinh}\left({t}\right)\:−\:{cosh}\left({t}\right)\right)^{\mathrm{2}} }=\mathrm{0} \\ $$$$\left.\mathrm{3}\right)\:{x}<\mathrm{0} \\ $$$$\int_{\mathrm{0}} ^{\:{x}<\mathrm{0}} {f}\left({t}\right){dt}=−\int_{{x}<\mathrm{0}} ^{\:\mathrm{0}} {f}\left({t}\right){dt}<\mathrm{0} \\ $$$${since}\:{f}\left({t}\right)>\mathrm{0}\:{for}\:{t}<\mathrm{0}\left({t}\neq\mathrm{0}\right) \\ $$$$\Rightarrow\nexists{x}<\mathrm{0},\:{s}.{t}.\:\int_{\mathrm{0}} ^{\:{x}} \frac{{t}^{\mathrm{2}} \:{dt}}{\left({t}\:{sinh}\left({t}\right)\:−\:{cosh}\left({t}\right)\right)^{\mathrm{2}} }=\mathrm{0} \\ $$$$ \\ $$$$ \\ $$$$\Rightarrow{Only}\:{soln}.\:{is}\:{x}=\mathrm{0} \\ $$

Commented by Shrinava last updated on 12/Jul/22

Perfect professor, thank you so much

$$\mathrm{Perfect}\:\mathrm{professor},\:\mathrm{thank}\:\mathrm{you}\:\mathrm{so}\:\mathrm{much} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com