Algebra Questions

Question Number 173418 by Shrinava last updated on 11/Jul/22

Answered by aleks041103 last updated on 12/Jul/22

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

Commented by Shrinava last updated on 12/Jul/22

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