Question Number 201859 by York12 last updated on 14/Dec/23 $$\mathrm{If}\:{xyz}\:\in\mathbb{R}^{+} \:,\:{xyz}=\mathrm{1}\:,\:\mathrm{prove}\:\mathrm{that}\:\mathrm{the}\:\mathrm{following}\:\mathrm{inequality}\:\mathrm{holds}: \\ $$$$\frac{{x}}{\mathrm{2}{x}^{\mathrm{5}} +{x}+\mathrm{4}}+\frac{{y}}{\mathrm{2}{y}^{\mathrm{5}} +{y}+\mathrm{4}}+\frac{{z}}{\mathrm{2}{z}^{\mathrm{5}} +{z}+\mathrm{4}}\geqslant\frac{\mathrm{3}}{\mathrm{7}}. \\ $$$$\boldsymbol{\mathrm{Solution}}\:\boldsymbol{\mathrm{please}}\:\boldsymbol{\mathrm{with}}\:\boldsymbol{\mathrm{an}}\:\boldsymbol{\mathrm{advice}}\:\boldsymbol{\mathrm{to}}\:\boldsymbol{\mathrm{get}}\:\boldsymbol{\mathrm{better}} \\ $$$$\boldsymbol{\mathrm{at}}\:\boldsymbol{\mathrm{inequalities}}\:\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{which}}\:\boldsymbol{\mathrm{book}}\:\boldsymbol{\mathrm{would}}\:\boldsymbol{\mathrm{you}}\:\boldsymbol{\mathrm{recommend}}. \\ $$$$\boldsymbol{\mathrm{Thanks}}\:\boldsymbol{\mathrm{in}}\:\boldsymbol{\mathrm{advance}}! \\ $$$$\: \\…
Question Number 201820 by cortano12 last updated on 13/Dec/23 $$\:\:\:\frac{\mid\mathrm{3x}+\mathrm{1}\mid−\mid\mathrm{x}+\mathrm{2}\mid}{\mathrm{3}−\mid\mathrm{2x}\mid}\:\geqslant\:\mathrm{0}\: \\ $$$$\:\:\mathrm{find}\:\mathrm{the}\:\mathrm{solution}\:\mathrm{set}. \\ $$ Answered by dimentri last updated on 13/Dec/23 $$\:\:\frac{\left(\mathrm{3x}+\mathrm{1}\right)^{\mathrm{2}} −\left(\mathrm{x}+\mathrm{2}\right)^{\mathrm{2}} }{\mathrm{3}^{\mathrm{2}} −\left(\mathrm{2x}\right)^{\mathrm{2}}…
Question Number 201837 by MrGHK last updated on 13/Dec/23 $${y}'''−{y}''+{y}'={sec}\left({t}\right),−\frac{\pi}{\mathrm{2}}<{t}<\frac{\pi}{\mathrm{2}} \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 201822 by MathedUp last updated on 13/Dec/23 $$\mathrm{Do}\:\mathrm{Not}\:\mathrm{Use}\:\mathrm{sin}\left(\theta\right)\sim\theta\:\left(\theta\:\:\mathrm{is}\:\mathrm{small}\:\mathrm{Enough}\right) \\ $$$$\ddot {\theta}+\frac{\mathrm{g}}{\ell}\mathrm{sin}\left(\theta\right)=\mathrm{0} \\ $$$${y}''\left({t}\right)+\frac{\mathrm{g}}{\ell}\:\mathrm{sin}\left({y}\left({t}\right)\right)=\mathrm{0} \\ $$$${y}''\left({t}\right){y}'\left({t}\right)+\frac{\mathrm{g}}{\ell}\mathrm{sin}\left({y}\left({t}\right)\right){y}'\left({t}\right)=\mathrm{0} \\ $$$${y}'\left({t}\right){y}''\left({t}\right)=\frac{\mathrm{1}}{\mathrm{2}}\centerdot\frac{\mathrm{d}\:\:}{\mathrm{d}{t}}\left({y}'\left({t}\right)\right)^{\mathrm{2}} \\ $$$$\frac{\mathrm{g}}{\ell}\mathrm{sin}\left({y}\left({t}\right)\right){y}'\left({t}\right)=−\frac{\mathrm{g}}{\ell}\centerdot\frac{\mathrm{d}\:\:}{\mathrm{d}{t}}\mathrm{cos}\left({y}\left({t}\right)\right) \\ $$$$\therefore\:\frac{\mathrm{d}\:\:}{\mathrm{d}{t}}\left[\frac{\mathrm{1}}{\mathrm{2}}\left({y}'\left({t}\right)\right)^{\mathrm{2}} −\frac{\mathrm{g}}{\ell}\mathrm{cos}\left({y}\left({t}\right)\right)\right]=\mathrm{0} \\…
Question Number 201832 by sonukgindia last updated on 13/Dec/23 Answered by aleks041103 last updated on 13/Dec/23 $${x}={e}^{−{t}} \:\Rightarrow\:{dx}=−{e}^{−{t}} {dt} \\ $$$$\Rightarrow{I}=\mathrm{32}\int_{\mathrm{0}} ^{\:\infty} \frac{{sin}^{\mathrm{5}} \left(−{t}\right)}{−{t}}{e}^{−{t}} {dt}=\mathrm{32}\int_{\mathrm{0}}…
Question Number 201833 by sebastian last updated on 13/Dec/23 $$ \\ $$$$ \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 201817 by Calculusboy last updated on 13/Dec/23 Answered by witcher3 last updated on 13/Dec/23 $$=\int\left(\mathrm{sin}\left(\mathrm{ln}\left(\mathrm{x}+\mathrm{1}\right)\right)\left(\sqrt{\mathrm{x}+\mathrm{1}}−\mathrm{1}\right)\right)^{\mathrm{2}} \mathrm{dx} \\ $$$$=\int\mathrm{sin}^{\mathrm{2}} \left(\mathrm{ln}\left(\mathrm{x}+\mathrm{1}\right)\right)\left(\mathrm{x}+\mathrm{2}−\mathrm{2}\sqrt{\mathrm{x}+\mathrm{1}}\right) \\ $$$$\mathrm{ln}\left(\sqrt{\mathrm{x}+\mathrm{1}}\right)=\mathrm{t} \\ $$$$\mathrm{x}+\mathrm{1}=\mathrm{e}^{\mathrm{2t}}…
Question Number 201834 by sebastian last updated on 13/Dec/23 $$ \\ $$$$ \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 201819 by York12 last updated on 13/Dec/23 $$\mathrm{If}\:{xyz}\:\in\mathbb{R}^{+} \:,\:{xyz}=\mathrm{1}\:,\:\mathrm{prove}\:\mathrm{that}\:\mathrm{the}\:\mathrm{following}\:\mathrm{inequality}\:\mathrm{holds}: \\ $$$$\frac{{x}}{\mathrm{2}{x}^{\mathrm{5}} +{x}+\mathrm{4}}+\frac{{y}}{\mathrm{2}{y}^{\mathrm{5}} +{y}+\mathrm{4}}+\frac{{z}}{\mathrm{2}{z}^{\mathrm{5}} +{z}+\mathrm{4}}\geqslant\frac{\mathrm{3}}{\mathrm{7}}. \\ $$$$\boldsymbol{\mathrm{Solution}}\:\boldsymbol{\mathrm{please}}\:\boldsymbol{\mathrm{with}}\:\boldsymbol{\mathrm{an}}\:\boldsymbol{\mathrm{advice}}\:\boldsymbol{\mathrm{to}}\:\boldsymbol{\mathrm{get}}\:\boldsymbol{\mathrm{better}} \\ $$$$\boldsymbol{\mathrm{at}}\:\boldsymbol{\mathrm{inequalities}}\:\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{which}}\:\boldsymbol{\mathrm{book}}\:\boldsymbol{\mathrm{would}}\:\boldsymbol{\mathrm{you}}\:\boldsymbol{\mathrm{recommend}}. \\ $$$$\boldsymbol{\mathrm{Thanks}}\:\boldsymbol{\mathrm{in}}\:\boldsymbol{\mathrm{advance}}! \\ $$$$\: \\…
Question Number 201829 by 281981 last updated on 13/Dec/23 $${shortest}\:{distance}\:{from}\:\left(−\mathrm{6},\mathrm{0}\right){to}\:{x}^{\mathrm{2}} −{y}^{\mathrm{2}} +\mathrm{16}=\mathrm{0} \\ $$ Answered by esmaeil last updated on 13/Dec/23 $${d}=\sqrt{\left({x}+\mathrm{6}\right)^{\mathrm{2}} +{y}^{\mathrm{2}} }=\sqrt{\overset{\mathrm{2}} {{x}}+\mathrm{12}{x}+\mathrm{36}+\overset{\mathrm{2}}…