Question-173418

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

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

Facebook Tweet Pin

Leave a Reply Cancel reply