Question Number 66759 by aliesam last updated on 19/Aug/19 Commented by mr W last updated on 19/Aug/19 $${question}\:{is}\:{wrong}. \\ $$$${there}\:{is}\:{no}\:{x}\:{which}\:{fulfills}\:\mid\frac{\mathrm{sin}\:{x}}{{x}^{\mathrm{2}} −{x}+\mathrm{1}}\mid>\mathrm{2}, \\ $$$${but}\:{such}\:{that}\:{e}^{{x}} >\mathrm{2016},\:{x}>\mathrm{ln}\:\mathrm{2016}\approx\mathrm{7}.\mathrm{6} \\…
Question Number 132295 by shaker last updated on 13/Feb/21 Commented by mathmax by abdo last updated on 13/Feb/21 $$\mathrm{f}\left(\mathrm{x}\right)=\mathrm{logx}−\sqrt{\mathrm{5x}−\mathrm{5}}\:\:\:\mathrm{f}\:\mathrm{defined}\:\mathrm{on}\:\left[\mathrm{1},+\infty\left[\right.\right. \\ $$$$\mathrm{f}^{'} \left(\mathrm{x}\right)=\frac{\mathrm{1}}{\mathrm{x}}−\frac{\mathrm{5}}{\mathrm{2}\sqrt{\mathrm{5x}−\mathrm{5}}}\:=\frac{\mathrm{1}}{\mathrm{x}}−\frac{\sqrt{\mathrm{5}}}{\mathrm{2}\sqrt{\mathrm{x}−\mathrm{1}}}\:=\frac{\mathrm{2}\sqrt{\mathrm{x}−\mathrm{1}}−\sqrt{\mathrm{5}}\mathrm{x}}{\mathrm{2x}\sqrt{\mathrm{x}−\mathrm{1}}} \\ $$$$=\frac{\left(\mathrm{2}\sqrt{\mathrm{x}−\mathrm{1}}−\sqrt{\mathrm{5}}\mathrm{x}\right)\left(\mathrm{2}\sqrt{\mathrm{x}−\mathrm{1}}+\sqrt{\mathrm{5}}\mathrm{x}\right)}{\mathrm{2x}\sqrt{\mathrm{x}−\mathrm{1}}\left(\mathrm{2}\sqrt{\mathrm{x}−\mathrm{1}}+\sqrt{\mathrm{5}}\mathrm{x}\right)}\:=\frac{\mathrm{4}\left(\mathrm{x}−\mathrm{1}\right)−\mathrm{5x}^{\mathrm{2}} }{\left(…\right)}=\frac{−\mathrm{5x}^{\mathrm{2}}…
Question Number 66756 by John Kaloki Musau last updated on 19/Aug/19 $${solve}\:{the}\:{equations}, \\ $$$${x}+{y}=\mathrm{17} \\ $$$${xy}−\mathrm{5}{x}=\mathrm{32} \\ $$ Answered by $@ty@m123 last updated on 19/Aug/19…
Question Number 132294 by Algoritm last updated on 13/Feb/21 Commented by mr W last updated on 13/Feb/21 $${x}={y}=\mathrm{1} \\ $$$${the}\:{proof}\:{you}\:{may}\:{ask}\:{for}\:{is}\:{thinking}. \\ $$ Terms of Service…
Question Number 66755 by John Kaloki Musau last updated on 19/Aug/19 $${solve}\:{the}\:{simultaneous}\:{equations}: \\ $$$$\mathrm{3}{x}-{y}=\mathrm{9} \\ $$$${x}^{\mathrm{2}} -{xy}=\mathrm{4} \\ $$ Answered by $@ty@m123 last updated on…
Question Number 132285 by bemath last updated on 13/Feb/21 $$\mathrm{Simplify}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{of}\: \\ $$$$\frac{\left(\mathrm{x}^{\frac{\mathrm{1}}{\mathrm{3}}} −\mathrm{x}^{\frac{\mathrm{1}}{\mathrm{6}}} \right)\left(\mathrm{x}^{\frac{\mathrm{1}}{\mathrm{2}}} +\mathrm{x}\right)\left(\mathrm{x}^{\frac{\mathrm{1}}{\mathrm{2}}} +\mathrm{x}^{\frac{\mathrm{1}}{\mathrm{3}}} +\mathrm{x}^{\frac{\mathrm{2}}{\mathrm{3}}} \right)}{\left(\mathrm{x}^{\frac{\mathrm{4}}{\mathrm{3}}} −\mathrm{x}\right)\left(\mathrm{x}+\mathrm{x}^{\frac{\mathrm{1}}{\mathrm{3}}} +\mathrm{x}^{\frac{\mathrm{2}}{\mathrm{3}}} \right)} \\ $$$$\mathrm{with}\:\mathrm{x}\:\neq\:\mathrm{0} \\ $$$$…
Question Number 66751 by John Kaloki Musau last updated on 19/Aug/19 $${A}\:{rostum}\:{is}\:{made}\:{by}\:{cutting}\:{off} \\ $$$${the}\:{upper}\:{part}\:{of}\:{a}\:{cone}\:{along}\:{a} \\ $$$${plane}\:{parallel}\:{to}\:{the}\:{base}\:{at}\:\frac{\mathrm{2}}{\mathrm{3}}\:{up}\: \\ $$$${the}\:{height}.\:{What}\:{fraction}\:{of}\:{the}\: \\ $$$${volume}\:{of}\:{the}\:{cone}\:{does}\:{the} \\ $$$${rostum}\:{represent}? \\ $$ Answered…
Question Number 132287 by benjo_mathlover last updated on 13/Feb/21 Answered by Olaf last updated on 13/Feb/21 $$\mathrm{Let}\:{q}\:=\:\mathrm{2}\int_{\mathrm{0}} ^{\mathrm{6}} {f}\left({x}\right){dx}\:=\:\mathrm{2}\int_{\mathrm{0}} ^{\mathrm{6}} {f}\left({x}−\mathrm{4}\right){dx}\:\:\:\:\left(\mathrm{1}\right) \\ $$$$\mathrm{Let}\:{u}\:=\:{x}−\mathrm{4} \\ $$$$\left(\mathrm{1}\right)\::\:{q}\:=\:\mathrm{2}\int_{−\mathrm{4}}…
Question Number 66746 by John Kaloki Musau last updated on 20/Aug/19 $${A}\:{point}\:{T}\:\:{divides}\:{a}\:{line}\:{AB}\:{internally}\:{in}\:{the}\:{ratio}\:\mathrm{5}:\mathrm{2}.\:{Given}\:{that}\:{A}\:{is}\:\left(-\mathrm{4},\mathrm{10}\right)\:{and}\:{B}\:{is}\:\left(\mathrm{10},\mathrm{3}\right),\:{find}\:{the}\:{coordinates}\:{of}\:{T}. \\ $$ Answered by mr W last updated on 19/Aug/19 $${x}_{{T}} ={x}_{{A}} +\frac{\mathrm{5}}{\mathrm{7}}\left({x}_{{B}}…
Question Number 1211 by 112358 last updated on 14/Jul/15 $${Is}\:{there}\:{a}\:{solution}\:{of}\:{y}\:{in}\:{terms} \\ $$$${of}\:{x}\:{for}\:{the}\:{following}\:{D}.{E}? \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\frac{{dy}}{{dx}}+\frac{{c}_{\mathrm{1}} }{{y}\left({c}_{\mathrm{2}} {x}+{c}_{\mathrm{3}} \right)^{\mathrm{2}} }={c}_{\mathrm{4}} \\ $$$${Here}\:{c}_{\mathrm{1}} ,\:{c}_{\mathrm{2}} ,\:{c}_{\mathrm{3}} ,\:{c}_{\mathrm{4}} \:{are}\:{constants}.\: \\…