Question Number 66759 by aliesam last updated on 19/Aug/19
Commented by mr W last updated on 19/Aug/19
$${question}\:{is}\:{wrong}. \\ $$$${there}\:{is}\:{no}\:{x}\:{which}\:{fulfills}\:\mid\frac{\mathrm{sin}\:{x}}{{x}^{\mathrm{2}} −{x}+\mathrm{1}}\mid>\mathrm{2}, \\ $$$${but}\:{such}\:{that}\:{e}^{{x}} >\mathrm{2016},\:{x}>\mathrm{ln}\:\mathrm{2016}\approx\mathrm{7}.\mathrm{6} \\ $$
Answered by mind is power last updated on 19/Aug/19
$$\mid\frac{{sin}\left({x}\right)}{{x}^{\mathrm{2}} −{x}+\mathrm{1}}\mid>\mathrm{2}<===>\mid{sin}\left({x}\right)\mid>\mathrm{2}\mid{x}^{\mathrm{2}} −{x}+\mathrm{1}\mid \\ $$$$\mid{x}^{\mathrm{2}} −{x}+\mathrm{1}\mid={x}^{\mathrm{2}} −{x}+\mathrm{1}=\left({x}−\frac{\mathrm{1}}{\mathrm{2}}\right)+\frac{\mathrm{3}}{\mathrm{4}}\geqslant\frac{\mathrm{3}}{\mathrm{4}} \\ $$$$==>\mid{sin}\left({x}\right)\mid>\mathrm{2}×\frac{\mathrm{3}}{\mathrm{4}}=\frac{\mathrm{3}}{\mathrm{2}}>\mathrm{1} \\ $$$${but}\:\forall{x}\in{IR}\:\:\:\mid{sin}\left({x}\right)\mid\leqslant\mathrm{1} \\ $$$${hens}\:\mid\frac{{sin}\left({x}\right)}{{x}^{\mathrm{2}} −{x}+\mathrm{1}}\mid>\mathrm{2}\:\:{isfalse} \\ $$$$\mid\frac{{sin}\left({x}\right)}{{x}^{\mathrm{2}} −{x}+\mathrm{1}}\mid>\mathrm{2}===>{e}^{{x}} >\mathrm{2016}\:\:\:\:\:{is}\:{true}\: \\ $$$${because}\:{A}===>{B}\:<===>={not}\:{A}\:{or}\:{B} \\ $$$$ \\ $$
Commented by mr W last updated on 19/Aug/19
$${sir},\:{how}\:{can}\:{you}\:{say}\:{if}\:\mid\frac{{sin}\left({x}\right)}{{x}^{\mathrm{2}} −{x}+\mathrm{1}}\mid>\mathrm{2} \\ $$$${is}\:{false},\:{then}\:{e}^{{x}} >\mathrm{2016}\:{is}\:{true}? \\ $$$${this}\:{is}\:{to}\:{say}\:{if}\:{statement}\:\mathrm{1}\:{is}\:{false}, \\ $$$${then}\:{statement}\:\mathrm{2}\:{is}\:\left({always}\right)\:{true}. \\ $$$${this}\:{is}\:{to}\:{say}\:{if}\:{the}\:{earth}\:{is}\:{larger}\:{than} \\ $$$${the}\:{sun},\:{then}\:{it}\:{will}\:{rain}\:{tomorrow}. \\ $$