Question Number 219890 by SdC355 last updated on 03/May/25 $$\mathrm{Find}\:\mathrm{Maxima}\: \\ $$$${x}+{y}\:\mathrm{where}\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} ={r}^{\mathrm{2}} \:\left(\mathrm{use}\:\mathrm{Lagrange}\:\mathrm{Method}\right) \\ $$ Answered by MrGaster last updated on 03/May/25 $$\bigtriangledown\left({x}+{y}\right)=\lambda\bigtriangledown\left({x}^{\mathrm{2}}…
Question Number 219884 by Rojarani last updated on 03/May/25 $$\:\left({a},{b},{c}\right)>\mathrm{0}\:{such}\:{that}, \\ $$$$\:\:{a}+{b}+{c}=\mathrm{13},\:\:{abc}=\mathrm{36} \\ $$$$\:\:{find}\:{the}\:{maximum}\:{and}\:{minimum}\: \\ $$$$\:{value}\:{of}\:\:{ab}+{bc}+{ca}=? \\ $$ Answered by MrGaster last updated on 03/May/25…
Question Number 219887 by SdC355 last updated on 03/May/25 $$\mathrm{what}\:\mathrm{is}\: \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\left(\mathrm{1}+\frac{\mathrm{1}}{{n}}\right)\uparrow\uparrow^{\infty} =?? \\ $$$${a}\uparrow\uparrow^{{m}} =\underset{{m}\:\mathrm{times}} {\underbrace{{a}^{{a}^{{a}^{{a}^{\iddots} } } } }}\:\:\left(\mathrm{aka}\:\mathrm{Knuth}'\mathrm{s}\:\mathrm{up}\:\mathrm{notation}\right) \\ $$ Answered…
Question Number 219940 by hardmath last updated on 03/May/25 $$\mathrm{Let}: \\ $$$$\mathrm{f}\::\:\left[\mathrm{n}−\mathrm{1}\:,\:\mathrm{n}\right]\:\rightarrow\:\left[\mathrm{n}\:,\:\mathrm{n}\:+\:\mathrm{1}\right] \\ $$$$\mathrm{be}\:\mathrm{a}\:\mathrm{continuous}\:\mathrm{function} \\ $$$$\mathrm{Such}\:\mathrm{that}: \\ $$$$\int_{\boldsymbol{\mathrm{n}}−\mathrm{1}} ^{\:\boldsymbol{\mathrm{n}}} \left(\mathrm{1}\:+\:\mathrm{xf}\:^{'} \left(\mathrm{x}\right)\right)\mathrm{dx}\:\leqslant\:\mathrm{nf}\left(\mathrm{n}\right)−\left(\mathrm{n}−\mathrm{1}\right)\mathrm{f}\left(\mathrm{n}−\mathrm{1}\right) \\ $$$$\mathrm{Then}\:\mathrm{prove}: \\ $$$$\int_{\boldsymbol{\mathrm{n}}−\mathrm{1}}…
Question Number 219879 by fantastic last updated on 03/May/25 Answered by A5T last updated on 03/May/25 $$\mathrm{Let}\:\mathrm{the}\:\mathrm{sides}\:\mathrm{of}\:\mathrm{the}\:\mathrm{rectngle}\:\mathrm{be}\:\mathrm{a}\:\mathrm{and}\:\mathrm{b}. \\ $$$$\sqrt{\left(\mathrm{1}.\mathrm{5}+\mathrm{2}\right)^{\mathrm{2}} −\left(\mathrm{a}−\mathrm{1}.\mathrm{5}−\mathrm{2}\right)^{\mathrm{2}} }+\sqrt{\left(\mathrm{3}+\mathrm{2}\right)^{\mathrm{2}} −\left(\mathrm{a}−\mathrm{3}−\mathrm{2}\right)^{\mathrm{2}} } \\ $$$$=\mathrm{b}−\mathrm{1}.\mathrm{5}−\mathrm{3}…
Question Number 219936 by Razafitiana last updated on 04/May/25 $$\mathrm{Prove}\:\mathrm{that}:\forall\mathrm{n}\in\mathrm{IN} \\ $$$$\underset{\:\mathrm{n}} {\int}^{\:\mathrm{n}+\mathrm{1}} \mathrm{ln}\left(\mathrm{t}\right)\mathrm{dt}\leqslant\mathrm{ln}\left(\mathrm{n}+\frac{\mathrm{1}}{\mathrm{2}}\right) \\ $$ Answered by MrGaster last updated on 04/May/25 $$\underset{\:\mathrm{n}} {\int}^{\:\mathrm{n}+\mathrm{1}}…
Question Number 219872 by SdC355 last updated on 03/May/25 $$\mathrm{prove} \\ $$$$\int\:\:{Y}_{−\frac{\mathrm{3}}{\mathrm{2}}} \left({z}\right)\:\mathrm{d}{z}=\frac{\mathrm{4sin}\left({z}\right)+\frac{{z}\boldsymbol{\Gamma}\left(\frac{\mathrm{1}}{\mathrm{2}},−\boldsymbol{{i}}{z}\right)}{\:\sqrt{−\boldsymbol{{i}}{z}}}+\frac{{z}\boldsymbol{\Gamma}\left(\frac{\mathrm{1}}{\mathrm{2}},\boldsymbol{{i}}{z}\right)}{\:\sqrt{\boldsymbol{{i}}{z}}}}{\:\sqrt{\mathrm{2}\pi{z}}}+{C} \\ $$ Answered by MrGaster last updated on 03/May/25 $${Y}_{−\nu} =\left(−\mathrm{1}\right)^{\nu} {Y}_{\nu}…
Question Number 219874 by mr W last updated on 03/May/25 $${find}\:\sqrt{\mathrm{2}^{\mathrm{6}^{\mathrm{2}^{\mathrm{1}^{\mathrm{4}^{\mathrm{4}} } } } } }=? \\ $$ Answered by fantastic last updated on 03/May/25…
Question Number 219933 by BaliramKumar last updated on 03/May/25 $$\mathrm{A}^{\mathrm{1}} \:+\:\mathrm{B}^{\mathrm{2}} \:+\:\mathrm{C}^{\mathrm{3}} \:+\:\mathrm{D}^{\mathrm{4}} \:=\:\overline {\mathrm{ABCD}} \\ $$$${find}\:\:{ABCD} \\ $$ Answered by MrGaster last updated on…
Question Number 219870 by SdC355 last updated on 03/May/25 $$\int_{\mathrm{0}} ^{\:\infty} \:{K}_{\nu} \left({r}\right)\mathrm{d}{r} \\ $$$$\int_{\mathrm{0}} ^{\:\infty} \:{t}\centerdot{Y}_{\mathrm{0}} \left({t}\right)\mathrm{d}{t} \\ $$$$\int_{\mathrm{0}} ^{\:\infty} \:\:\frac{\mathrm{sin}\left({t}\right){e}^{−{kt}} }{{t}^{\mathrm{2}} +\rho^{\mathrm{2}} }\mathrm{d}{t}\:…