Question Number 170155 by cortano1 last updated on 17/May/22 $$\:\:{Given}\:{f}\left({x}\right)={x}\sqrt{\mathrm{1}−{x}+\sqrt{\mathrm{1}−{x}}} \\ $$$$\:{where}\:\mathrm{0}\leqslant{x}\leqslant\mathrm{1} \\ $$$$\:{find}\:{max}\:{f}\left({x}\right) \\ $$ Answered by Mathspace last updated on 18/May/22 $${x}\in\left[\mathrm{0},\mathrm{1}\right]\:\Rightarrow{x}={cos}\theta\:\Rightarrow \\…
Question Number 104608 by ~blr237~ last updated on 22/Jul/20 $$\:\:{D}^{\frac{\mathrm{1}}{\mathrm{2}}} \left({y}\right)=\mathrm{1}\:\:\:\:\:\Rightarrow\:\:\:{y}\left({x}\right)=\sqrt{\frac{{x}}{\pi}} \\ $$ Answered by OlafThorendsen last updated on 22/Jul/20 $${n}\in\mathbb{N},\:\nu\in\mathbb{C},\:\left({x}^{\nu} \right)^{\left({n}\right)} \:=\:\frac{\Gamma\left(\nu+\mathrm{1}\right)}{\Gamma\left(\nu+\mathrm{1}−{n}\right)}{x}^{\nu−{n}} \\ $$$$\mathrm{By}\:\mathrm{generalizing}\::…
Question Number 104604 by ~blr237~ last updated on 22/Jul/20 $$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}}+…+\frac{\mathrm{1}}{{n}}\right)\frac{\mathrm{1}}{\mathrm{2}^{{n}} }\:\:=\:{ln}\mathrm{4} \\ $$ Answered by OlafThorendsen last updated on 22/Jul/20 $$\mathrm{S}\:=\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}}+…+\frac{\mathrm{1}}{{n}}\right)\frac{\mathrm{1}}{\mathrm{2}^{{n}}…
Question Number 38905 by Raj Singh last updated on 01/Jul/18 Answered by MrW3 last updated on 01/Jul/18 $$\frac{{a}}{\mathrm{sin}\:{A}}=\frac{{b}}{\mathrm{sin}\:{B}}=\frac{{c}}{\mathrm{sin}\:{C}}={k}\:\left({constant}\right) \\ $$$${let}\:{dA}\:={little}\:{change}\:{in}\:{angle}\:{A} \\ $$$$ \\ $$$${B}=\mathrm{180}−{C}−{A} \\…
Question Number 104302 by bobhans last updated on 20/Jul/20 Commented by bobhans last updated on 21/Jul/20 $${by}\:{parts} \\ $$$$\begin{cases}{{u}=\mathrm{arc}\:\mathrm{tan}\:\left(\frac{\mathrm{1}}{\mathrm{2}{x}^{\mathrm{2}} }\right)\:\Rightarrow{du}\:=\frac{−\frac{\mathrm{1}}{{x}^{\mathrm{3}} }}{\mathrm{1}+\frac{\mathrm{1}}{\mathrm{4}{x}^{\mathrm{4}} }}\:{dx}}\\{{dv}\:=\:{dx}\:\Rightarrow{v}\:=\:{x}}\end{cases} \\ $$$${I}=\:{x}\:\mathrm{arc}\:\mathrm{tan}\:\left(\frac{\mathrm{1}}{\mathrm{2}{x}^{\mathrm{2}} }\right)+\int\:\frac{\mathrm{4}{x}^{\mathrm{2}}…
Question Number 38762 by Raj Singh last updated on 29/Jun/18 Commented by tanmay.chaudhury50@gmail.com last updated on 29/Jun/18 Commented by abdo.msup.com last updated on 29/Jun/18 $${we}\:{have}\:{f}\left({x}\right)={arctan}\left({cosx}\:+{sinx}\right)…
Question Number 169826 by cortano1 last updated on 10/May/22 $$\:\:{Given}\:{f}\left({x}\right)=\sqrt{\mathrm{sin}\:{x}}\:+\sqrt{\mathrm{3}\:\mathrm{cos}\:{x}}\: \\ $$$$\:\:{x}\in\:\left(\mathrm{0},\:\frac{\pi}{\mathrm{2}}\right) \\ $$$$\:{Find}\:{max}\:{f}\left({x}\right).\: \\ $$ Answered by mr W last updated on 10/May/22 $${f}'\left({x}\right)=\frac{\mathrm{cos}\:{x}}{\mathrm{2}\sqrt{\mathrm{sin}\:{x}}}−\frac{\mathrm{3sin}\:{x}}{\mathrm{2}\sqrt{\mathrm{3}\:\mathrm{cos}\:{x}}}=\mathrm{0}…
Question Number 169553 by ali009 last updated on 02/May/22 $${solve}\:{the}\:{D}.{E} \\ $$$$\mathrm{2}{dx}−{e}^{{y}−{x}} {dy}=\mathrm{0} \\ $$ Answered by som(math1967) last updated on 03/May/22 $$\:\mathrm{2}{dx}={e}^{{y}−{x}} {dy} \\…
Question Number 169549 by ali009 last updated on 02/May/22 $${convert}\:{this}\:{D}.{E}\:{to}\:{exact}\:{D}.{E}\:{and}\:{solve}\:{it} \\ $$$${ydx}+{x}\left(\mathrm{1}+{y}\right){dy}=\mathrm{0} \\ $$ Answered by aleks041103 last updated on 02/May/22 $${y}+{x}\left(\mathrm{1}+{y}\right)\frac{{dy}}{{dx}}=\mathrm{0} \\ $$$$\Rightarrow\left(\mathrm{1}+{y}\right)\frac{{dy}}{{dx}}=−\frac{{y}}{{x}} \\…
Question Number 103998 by M±th+et+s last updated on 18/Jul/20 $${if}\:\:\:\:{f}\left({x}\right)={x}^{\frac{\mathrm{3}}{\mathrm{2}}} \\ $$$${f}'\left(\mathrm{0}\right)=\mathrm{0}\:\:{or}\:{not}\:{exist} \\ $$ Commented by M±th+et+s last updated on 19/Jul/20 Commented by M±th+et+s last…