Question Number 204621 by hardmath last updated on 23/Feb/24 $$\mathrm{a}\:,\:\mathrm{b}\:,\:\mathrm{c}\:\in\:\mathbb{R}^{+} \\ $$$$\mathrm{If}\:\:\:\sqrt{\mathrm{a}}\:+\:\sqrt{\mathrm{b}}\:+\:\sqrt{\mathrm{c}}\:=\:\mathrm{1} \\ $$$$\mathrm{Prove}\:\mathrm{that}:\:\:\:\mathrm{a}\:+\:\mathrm{b}\:+\:\mathrm{c}\:\geqslant\:\frac{\mathrm{1}}{\mathrm{3}} \\ $$ Answered by A5T last updated on 23/Feb/24 $$\frac{{a}+{b}+{c}}{\mathrm{3}}\geqslant\left(\frac{\sqrt{{a}}+\sqrt{{b}}+\sqrt{{c}}}{\mathrm{3}}\right)^{\mathrm{2}} =\frac{\mathrm{1}}{\mathrm{9}}\Rightarrow{a}+{b}+{c}\geqslant\frac{\mathrm{1}}{\mathrm{3}}…
Question Number 204617 by Abdullahrussell last updated on 23/Feb/24 Commented by Frix last updated on 23/Feb/24 $$\mathrm{33} \\ $$ Terms of Service Privacy Policy Contact:…
Question Number 204618 by es last updated on 23/Feb/24 $${if}\:\:\mathrm{7}{x}=\frac{\pi}{\mathrm{2}}\rightarrow\frac{{cosxsin}\mathrm{2}{xtan}\mathrm{3}{x}}{{cot}\mathrm{4}{xcos}\mathrm{5}{xsin}\mathrm{6}{x}}=? \\ $$ Answered by A5T last updated on 23/Feb/24 $$\frac{\frac{{cos}\left({x}\right){sin}\left(\mathrm{2}{x}\right){sin}\left(\mathrm{3}{x}\right)}{{cos}\left(\mathrm{3}{x}\right)}}{\frac{{cos}\left(\mathrm{4}{x}\right){cos}\left(\mathrm{5}{x}\right){sin}\left(\mathrm{6}{x}\right)}{{sin}\left(\mathrm{4}{x}\right)}} \\ $$$$=\frac{{sin}\left(\mathrm{4}{x}\right){cos}\left({x}\right){sin}\left(\mathrm{2}{x}\right){sin}\left(\mathrm{3}{x}\right)}{{cos}\left(\mathrm{3}{x}\right){cos}\left(\mathrm{4}{x}\right){cos}\left(\mathrm{5}{x}\right){sin}\left(\mathrm{6}{x}\right)}=\mathrm{1} \\ $$$$\left[{since}\:{sin}\left(\mathrm{4}{x}\right)={cos}\left(\mathrm{3}{x}\right);{cos}\left({x}\right)={sin}\left(\mathrm{6}{x}\right);\right. \\…
Question Number 204615 by lepuissantcedricjunior last updated on 26/Feb/24 $$\:\:\:\:\:\:\:\:\frac{\boldsymbol{\mathrm{exercice}}\:}{} \\ $$$$\:\:\:\:\:\:\:\:\boldsymbol{\mathrm{prouver}}\:\int_{\mathrm{0}} ^{\boldsymbol{\pi}} \int_{\mathrm{0}} ^{\boldsymbol{\mathrm{x}}} \boldsymbol{{sin}}\left(\frac{\boldsymbol{\mathrm{x}}^{\mathrm{2}} }{\boldsymbol{\pi}}\right)\boldsymbol{{d}\mathrm{x}{d}\mathrm{y}}=\boldsymbol{\pi} \\ $$$$\: \\ $$$$\:\:……………\boldsymbol{{prof}}\:\boldsymbol{{cedric}}\:\boldsymbol{{junior}}……….. \\ $$$$ \\ $$…
Question Number 204610 by mnjuly1970 last updated on 23/Feb/24 $$ \\ $$$$\:\:\:{If}\:,\:\:\:{f}\left({x}\right)\:=\:\begin{cases}{\:\mathrm{2}^{\mathrm{2}{x}} −\:{log}_{\mathrm{3}} \:\left(\:{x}+\mathrm{3}\:\right)\:\:\:\:;\:\:\:{x}\:\geqslant\mathrm{5}}\\{\:{f}\:\left(\mathrm{1}+\:{x}\:\right)\:\:−\mathrm{4}\:\:\:\:\:\:\:\:\:\:\:\:\:\:;\:\:{x}\:<\:\mathrm{5}}\end{cases}\:\:\:\:\: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\Rightarrow\:\:{f}\:\left(\mathrm{0}\:\right)=\:? \\ $$$$ \\ $$ Answered by Rasheed.Sindhi…
Question Number 204573 by MathedUp last updated on 22/Feb/24 $$\mathrm{How}\:\mathrm{Can}\:\mathrm{derive}\:\mathrm{LambertW}\left({z}\right)\:\mathrm{in}\:\mathrm{the} \\ $$$$\:\mathrm{Form}\:\mathrm{of}\:\mathrm{integral}??? \\ $$$$\mathrm{W}\left({z}\right)=\frac{\mathrm{1}}{\pi}\int_{\mathrm{0}} ^{\:\pi} \:\mathrm{ln}\left(\mathrm{1}+\frac{{z}\centerdot\mathrm{sin}\left({t}\right)}{{t}}{e}^{{t}\centerdot\mathrm{cot}\left({t}\right)} \right)\mathrm{d}{t}\:,\:{z}\in\left[−\frac{\mathrm{1}}{{e}},\infty\right) \\ $$$$\mathrm{Or}\:\mathrm{Similar}\:\mathrm{to}\:\mathrm{the}\:\mathrm{example}.\mathrm{LambertW}\left({z}\right) \\ $$$$\mathrm{How}\:\mathrm{other}\:\mathrm{Functions}\:\mathrm{can}\:\mathrm{be}\:\mathrm{Derived}\:\mathrm{in}\:\mathrm{Integral}\:\mathrm{Form} \\ $$ Commented by…
Question Number 204590 by sphelele last updated on 22/Feb/24 Answered by A5T last updated on 22/Feb/24 $$\left.{a}\right)=\frac{\mathrm{2}^{{n}+\mathrm{2}+\mathrm{2}{n}+\mathrm{2}} }{\mathrm{2}^{\mathrm{3}{n}−\mathrm{3}} }=\frac{\mathrm{2}^{\mathrm{3}{n}+\mathrm{4}} }{\mathrm{2}^{\mathrm{3}{n}+\mathrm{4}} \mathrm{2}^{−\mathrm{7}} }=\mathrm{128} \\ $$$$\left.{b}\right)=\left(\frac{{b}−{a}}{{ab}}\overset{−\mathrm{1}} {\right)}=\frac{{ab}}{{b}−{a}}…
Question Number 204574 by universe last updated on 22/Feb/24 $$ \\ $$$$\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\:\mathrm{n}^{−\mathrm{3}/\mathrm{2}} \left[\left(\mathrm{n}+\mathrm{1}\right)^{\left(\mathrm{n}+\mathrm{1}\right)} \left(\mathrm{n}+\mathrm{2}\right)^{\left(\mathrm{n}+\mathrm{2}\right)} …\left(\mathrm{2n}\right)^{\mathrm{2n}} \right]^{\mathrm{1}/\mathrm{n}^{\mathrm{2}} } \:=\:? \\ $$$$ \\ $$ Answered by…
Question Number 204598 by SEKRET last updated on 22/Feb/24 $$\:\:\:\:\:\:\:\:\int\sqrt[{\mathrm{3}}]{\boldsymbol{\mathrm{x}}−\boldsymbol{\mathrm{x}}^{\mathrm{2}} }\:\boldsymbol{\mathrm{dx}} \\ $$$$ \\ $$ Commented by Frix last updated on 23/Feb/24 $${t}=\mathrm{sin}^{−\mathrm{1}} \:\sqrt{{x}}\:\Rightarrow \\…
Question Number 204583 by hardmath last updated on 22/Feb/24 $$\mathrm{For}\:\:\:\mathrm{z}\:=\:\mathrm{a}\:−\:\mathrm{bi} \\ $$$$\mathrm{If}\:\:\:\left(\mid\mathrm{z}\mid\:−\:\mathrm{z}\right)\centerdot\left(\mid\mathrm{z}\mid\:+\:\overline {\mathrm{z}}\right)\:=\:\mathrm{4bi} \\ $$$$\mathrm{Find}\:\:\:\mid\mathrm{z}\mid\:=\:? \\ $$ Answered by A5T last updated on 22/Feb/24 $$\left(\mid{z}\mid−{z}\right)\left(\mid{z}\mid+\overset{−}…