Question Number 141136 by bobhans last updated on 16/May/21 $$\:\:\:\:{Find}\:{the}\:{smallest}\:{value}\: \\ $$$$\:\:\:\:\mathrm{5}{x}\:+\:\frac{\mathrm{16}}{{x}}\:+\:\mathrm{21}\:{over}\:{positive}\: \\ $$$$\:\:\:{value}\:{of}\:{x}\: \\ $$ Answered by iloveisrael last updated on 16/May/21 $$\:\Rightarrow\:\mathrm{5}{x}\:\&\:\frac{\mathrm{16}}{{x}}\:{have}\:{a}\:{constant}\:{product} \\…
Question Number 10064 by PradipGos. last updated on 22/Jan/17 $$\mathrm{if}\:\mathrm{a}^{\mathrm{2}} +\mathrm{b}^{\mathrm{2}} =\mathrm{3ab}\:\mathrm{then}\:\mathrm{prove}\:\mathrm{that}\: \\ $$$$\mathrm{log}\frac{\mathrm{a}+\mathrm{b}}{\:\sqrt{\mathrm{5}}}=\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{log}{a}+\mathrm{log}{b}\right) \\ $$$$ \\ $$ Answered by ridwan balatif last updated on…
Question Number 141133 by mnjuly1970 last updated on 16/May/21 $$\: \\ $$$$\:\:\:\:\:\:\:\:\:\boldsymbol{\mathrm{I}}:=\int_{\mathrm{0}} ^{\:\infty} \frac{\left({x}^{{n}} −\mathrm{1}\right)\left({x}−\mathrm{1}\right)}{{x}^{{n}+\mathrm{3}} −\mathrm{1}}{dx}=?? \\ $$$$ \\ $$ Answered by mathmax by abdo…
Question Number 75598 by A8;15: last updated on 13/Dec/19 Commented by mr W last updated on 13/Dec/19 $${please}\:{see}\:{Q}#\mathrm{74970} \\ $$ Terms of Service Privacy Policy…
Question Number 141132 by mnjuly1970 last updated on 16/May/21 $$\:\:\: \\ $$$$\:…{mathematical}\:…{analysis}… \\ $$$$\:\:\:\:{prove}\:\:{that}: \\ $$$$\:\:\:\:\:\:\:\:\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\prod}}\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{2}^{{n}} } }\:\right)\:\overset{?} {=}\:\mathrm{2}\:\:\: \\ $$$$\:\:\:\:\:\:…… \\ $$…
Question Number 10062 by ridwan balatif last updated on 22/Jan/17 Commented by ridwan balatif last updated on 22/Jan/17 $$\mathrm{find}\:\mathrm{the}\:\mathrm{remainder}\:\mathrm{of}\:\mathrm{the}\:\mathrm{division}\:\mathrm{20}^{\mathrm{2017}} +\mathrm{1}^{\mathrm{2017}} +\mathrm{17}^{\mathrm{2017}} +\mathrm{72}^{\mathrm{2017}} \:\mathrm{by}\:\mathrm{2017} \\ $$…
Question Number 141135 by bobhans last updated on 16/May/21 $$\:\:\:\:\:\:\:{Find}\:{the}\:{minimum}\:{of}\: \\ $$$$\:\:\:\:\:\:\frac{\mathrm{12}}{{x}}\:+\:\frac{\mathrm{18}}{{y}}\:+\:{xy}\:{for}\:{all}\: \\ $$$$\:\:\:\:\:\:{positive}\:{number}\:{x}\:\&\:{y}\:. \\ $$ Answered by mitica last updated on 16/May/21 $$\frac{\mathrm{12}}{{x}}+\frac{\mathrm{18}}{{y}}+{xy}\geqslant\mathrm{3}\sqrt[{\mathrm{3}}]{\frac{\mathrm{12}}{{x}}\centerdot\frac{\mathrm{18}}{{y}}\centerdot{xy}}=\mathrm{18} \\…
Question Number 141134 by mnjuly1970 last updated on 16/May/21 $$\:\:\:\:\: \\ $$$$\:\:\:{prove}\:{that}:: \\ $$$$\:\:\Phi:=\underset{{m},{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} }{{m}^{\mathrm{2}} +{n}^{\mathrm{2}} }=\:\frac{\pi^{\mathrm{2}} }{\mathrm{24}}+\frac{\pi{ln}\left(\mathrm{2}\right)}{\mathrm{8}} \\ $$$$ \\ $$ Answered…
Question Number 75597 by aliesam last updated on 13/Dec/19 Commented by mathmax by abdo last updated on 13/Dec/19 $${let}\:{f}\left({x}\right)=\frac{\left({cosx}\right)^{\frac{\mathrm{1}}{{m}}} −\left({cosx}\right)^{\frac{\mathrm{1}}{{n}}} }{{x}^{\mathrm{2}} }\:\:{we}\:{have} \\ $$$${cosx}\:\sim\mathrm{1}−\frac{{x}^{\mathrm{2}} }{\mathrm{2}}\:\Rightarrow\left({cosx}\right)^{\frac{\mathrm{1}}{{m}}}…
Question Number 141129 by bobhans last updated on 16/May/21 $${For}\:{what}\:{values}\:{of}\:\lambda\:{are}\:{the} \\ $$$${vectors}\:\lambda\hat {{i}}\:+\:\mathrm{2}\hat {{j}}\:+\:\hat {{k}}\:,\:\mathrm{3}\hat {{i}}\:+\mathrm{4}\hat {{j}}\:+\lambda\hat {{k}}\: \\ $$$$,\:\hat {{j}}\:+\:\hat {{k}}\:\:{coplanar}\: \\ $$ Answered…