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

Answered by aleks041103 last updated on 12/Jul/22

$$1)obvioussoln.x=0$$$$Observethat:$$$$\frac{t^2}{{(tsinh\left(t\right)-cosh\left(t\right))}^2}⩾0(equalityfort\ne 0)$$$$2)x>0$$$${\int }_0^{x>0}f\left(t\right)dt>0,sincef\left(t\right)>0fort>0.$$$$\Rightarrow \nexists x>0,s.t.{\int }_0^x\frac{t^{2}dt}{{(tsinh\left(t\right)-cosh\left(t\right))}^2}=0$$$$3)x<0$$$${\int }_0^{x<0}f\left(t\right)dt=-{\int }_{x<0}^{0}f\left(t\right)dt<0$$$$sincef\left(t\right)>0fort<0(t\ne 0)$$$$\Rightarrow \nexists x<0,s.t.{\int }_0^x\frac{t^{2}dt}{{(tsinh\left(t\right)-cosh\left(t\right))}^2}=0$$$$$$$$$$$$\Rightarrow Onlysoln.isx=0$$

Commented by Shrinava last updated on 12/Jul/22

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

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