Question Number 2454 by Yozzi last updated on 20/Nov/15 $${Find}\:{smallest}\:{a}>\mathrm{1}\:{for}\:{which} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{{a}+{sinx}}{{a}+{siny}}\leqslant{e}^{{y}−{x}} \\ $$$${for}\:\forall\:{x}\leqslant{y}. \\ $$ Commented by Rasheed Soomro last updated on 21/Nov/15 $${Find}\:{smallest}\:{a}>\mathrm{1}\:{for}\:{which}…
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Question Number 67983 by peter frank last updated on 03/Sep/19 Commented by Abdo msup. last updated on 03/Sep/19 $$\left.{b}\right)\:{let}\:{f}\left({x}\right)=\left(\frac{\mathrm{1}}{{x}}\right)^{{x}\:} \:\Rightarrow{f}\left({x}\right)={x}^{−{x}} \:={e}^{−{xln}\left({x}\right)} \\ $$$$\left.{f}\:{is}?{defined}\:{on}\:\right]\mathrm{0},+\infty\left[\right. \\ $$$${lim}_{{x}\rightarrow\mathrm{0}\:{and}\:{x}>\mathrm{0}}…
Question Number 133518 by Abdoulaye last updated on 22/Feb/21 Answered by mnjuly1970 last updated on 22/Feb/21 $${ans}:\:\frac{\pi^{\mathrm{3}} }{\mathrm{16}} \\ $$ Answered by Dwaipayan Shikari last…
Question Number 2441 by Yozzi last updated on 20/Nov/15 $${Prove}\:{or}\:{disprove}\:{that}\:,\:{for}\:{even}\: \\ $$$${positive}\:{n}, \\ $$$$\mathrm{2}×\underset{{k}=\mathrm{1}} {\overset{\frac{{n}}{\mathrm{2}}−\mathrm{1}} {\sum}}\left(−\mathrm{1}\right)^{{k}} \begin{pmatrix}{{n}}\\{{k}}\end{pmatrix}+\left(−\mathrm{1}\right)^{\left({n}/\mathrm{2}\right)} \frac{{n}!}{\left(\left({n}/\mathrm{2}\right)!\right)^{\mathrm{2}} }=−\mathrm{2} \\ $$ Commented by Rasheed Soomro…
Question Number 67977 by behi83417@gmail.com last updated on 02/Sep/19 Commented by behi83417@gmail.com last updated on 02/Sep/19 $$\boldsymbol{\mathrm{AD}}=\boldsymbol{\mathrm{DC}},\angle\boldsymbol{\mathrm{AEB}}=\mathrm{90}^{\bullet} \:\:. \\ $$$$\boldsymbol{\mathrm{find}}:\:\:\:\boldsymbol{\mathrm{ED}}\:\:\boldsymbol{\mathrm{in}}\:\boldsymbol{\mathrm{terms}}\:\boldsymbol{\mathrm{of}}:\boldsymbol{\mathrm{a}},\boldsymbol{\mathrm{b}},\boldsymbol{\mathrm{c}}. \\ $$ Answered by $@ty@m123…
Question Number 67974 by mathmax by abdo last updated on 02/Sep/19 $${let}\:{F}\left({x}\right)\:=\int_{{x}} ^{{x}^{\mathrm{2}} } \:\:\:\frac{{arctan}\left({xt}\right)}{{x}^{\mathrm{2}} \:+{t}^{\mathrm{2}} }{dt}\:\:{calculate}\:{F}\:^{'} \left({x}\right). \\ $$ Commented by mathmax by abdo…
Question Number 133508 by Ahmed1hamouda last updated on 22/Feb/21 Commented by Ahmed1hamouda last updated on 22/Feb/21 $$\boldsymbol{\mathrm{S}}\mathrm{olve}\:\mathrm{the}\:\mathrm{differential}\:\mathrm{equations} \\ $$ Answered by Olaf last updated on…
Question Number 67972 by mathmax by abdo last updated on 02/Sep/19 $${if}\:{F}\left({x}\right)=\int_{{u}\left({x}\right)} ^{{v}\left({x}\right)} {g}\left({x},{t}\right){dt}\:\:\:\:\:{determine}\:{a}\:{expression}\:{for}\:{F}\:^{'} \left({x}\right). \\ $$ Answered by Tanmay chaudhury last updated on 03/Sep/19…
Question Number 67973 by behi83417@gmail.com last updated on 02/Sep/19 Commented by behi83417@gmail.com last updated on 02/Sep/19 $$\mathrm{A}\overset{\bigtriangleup} {\mathrm{B}C},\mathrm{is}\:\mathrm{given}. \\ $$$$\boldsymbol{\mathrm{show}}\:\boldsymbol{\mathrm{that}}: \\ $$$$\mathrm{1}.\:\:\:\:\:\frac{\boldsymbol{\mathrm{BX}}}{\boldsymbol{\mathrm{XC}}}\:×\:\:\frac{\boldsymbol{\mathrm{CY}}}{\boldsymbol{\mathrm{YA}}}\:×\:\frac{\boldsymbol{\mathrm{AZ}}}{\boldsymbol{\mathrm{ZB}}}=\mathrm{1}. \\ $$$$\mathrm{2}.\:\:\:\:\:\frac{\boldsymbol{\mathrm{PX}}}{\boldsymbol{\mathrm{XA}}}\:+\:\:\frac{\boldsymbol{\mathrm{PY}}}{\boldsymbol{\mathrm{YB}}}\:+\:\frac{\boldsymbol{\mathrm{PZ}}}{\boldsymbol{\mathrm{ZC}}}=\mathrm{1}. \\…