Question Number 129731 by Adel last updated on 18/Jan/21 $$\underset{{x}\rightarrow\infty} {\mathrm{lim}cos}\:\frac{\mathrm{x}}{\mathrm{2}}×\mathrm{cos}\:\frac{\mathrm{x}}{\mathrm{4}}…….\mathrm{cos}\:\frac{\mathrm{x}}{\mathrm{2}^{\mathrm{n}} }=? \\ $$ Answered by Dwaipayan Shikari last updated on 18/Jan/21 $${cos}\frac{{x}}{\mathrm{2}}=\frac{{sinx}}{\mathrm{2}{sin}\frac{{x}}{\mathrm{2}}} \\ $$$$\underset{{n}=\mathrm{1}}…
Question Number 129727 by abdurehime last updated on 18/Jan/21 $$\mathrm{find}\:\:\:\:\frac{\mathrm{dy}}{\mathrm{dx}}\:\mathrm{if}\:\:\:\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} =\mathrm{4} \\ $$ Answered by Ar Brandon last updated on 18/Jan/21 $$\mathrm{y}=\pm\sqrt{\mathrm{4}−\mathrm{x}^{\mathrm{2}} } \\…
Question Number 64187 by Hope last updated on 15/Jul/19 Answered by Hope last updated on 15/Jul/19 $$\frac{{yz}}{{xyz}}+\frac{\mathrm{4}{zx}}{{xyz}}+\frac{\mathrm{9}{xy}}{{xyz}} \\ $$$$\frac{\mathrm{1}^{\mathrm{2}} }{{x}}+\frac{\mathrm{2}^{\mathrm{2}} }{{y}}+\frac{\mathrm{3}^{\mathrm{2}} }{{z}}\geqslant\frac{\left(\mathrm{1}+\mathrm{2}+\mathrm{3}\right)^{\mathrm{2}} }{\left({x}+{y}+{z}\right)} \\ $$$$\frac{\mathrm{1}^{\mathrm{2}}…
Question Number 129723 by Adel last updated on 18/Jan/21 $$\underset{{x}\rightarrow\mathrm{100}} {\mathrm{lim}}\left[\left(\mathrm{x}^{\mathrm{100}} −\mathrm{100}^{\mathrm{x}} \right)\frac{\mathrm{1}}{\mathrm{100}−\mathrm{x}}\right]=? \\ $$$$\mathcal{DANISH} \\ $$ Answered by bramlexs22 last updated on 18/Jan/21 $$\:\underset{{x}\rightarrow\mathrm{100}}…
Question Number 64186 by Hope last updated on 15/Jul/19 Answered by Hope last updated on 15/Jul/19 $$\frac{\left(\sqrt{\mathrm{2}}\:\right)^{\mathrm{2}} }{{x}+{y}}+\frac{\left(\sqrt{\mathrm{2}}\:\right)^{\mathrm{2}} }{{y}+{z}}+\frac{\left(\sqrt{\mathrm{2}}\:\right)^{\mathrm{2}} }{{z}+{x}}\geqslant\frac{\left(\sqrt{\mathrm{2}}\:+\sqrt{\mathrm{2}}\:+\sqrt{\mathrm{2}}\:\right)^{\mathrm{2}} }{\mathrm{2}\left({x}+{y}+{z}\right)} \\ $$$${do}\geqslant\frac{\mathrm{9}×\mathrm{2}}{\mathrm{2}\left({x}+{y}+{z}\right)} \\ $$$$\frac{\mathrm{2}}{{x}+{y}}+\frac{\mathrm{2}}{{y}+{z}}+\frac{\mathrm{2}}{{z}+{x}}\geqslant\frac{\mathrm{9}}{{x}+{y}+{z}}\:{proved}…
Question Number 64185 by Hope last updated on 15/Jul/19 Commented by Hope last updated on 15/Jul/19 $${all}\:{question}\:{based}\:{on}\:{Tito}−{lemma}\:{inequality} \\ $$ Commented by Tony Lin last updated…
Question Number 129719 by bramlexs22 last updated on 18/Jan/21 $$\:\:\mathrm{cos}\:\left(\pi\mathrm{sin}\:\left(\frac{\mathrm{x}}{\mathrm{3}}\right)\right)=\mathrm{sin}\:\left(\pi\mathrm{cos}\:\left(\frac{\mathrm{x}}{\mathrm{3}}\right)\right) \\ $$$$\:\:\mathrm{x}\:=? \\ $$$$ \\ $$ Answered by liberty last updated on 18/Jan/21 $$\:\mathrm{cos}\:\left(\pi\mathrm{sin}\:\frac{\mathrm{x}}{\mathrm{3}}\right)=\mathrm{cos}\:\left(\frac{\pi}{\mathrm{2}}−\pi\mathrm{cos}\:\frac{\mathrm{x}}{\mathrm{3}}\right) \\…
Question Number 129712 by liberty last updated on 18/Jan/21 Commented by bemath last updated on 18/Jan/21 $$\mathrm{edit}\:\mathrm{the}\:\mathrm{last}\:\mathrm{line}\equiv\:\frac{\mathrm{3}−\mathrm{0}+\frac{\mathrm{3}}{\mathrm{4}}−\frac{\mathrm{3}}{\mathrm{8}}}{\mathrm{1}−\mathrm{0}−\frac{\mathrm{3}}{\mathrm{4}}+\frac{\mathrm{1}}{\mathrm{8}}}=\mathrm{9} \\ $$ Commented by liberty last updated on…
Question Number 64176 by aliesam last updated on 15/Jul/19 Answered by MJS last updated on 15/Jul/19 $$\mathrm{the}\:\mathrm{minimum}\:\mathrm{of}\:{f}\left({x}\right)=\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }\:\mathrm{is}\:{f}\left(\mathrm{0}\right)=\mathrm{1}\:\Rightarrow\:\mathrm{the} \\ $$$$\mathrm{area}\:\mathrm{between}\:\mathrm{the}\:{x}−\mathrm{axis}\:\mathrm{and}\:{f}\left({x}\right)\:\mathrm{in}\:\left[−\mathrm{1};\:\mathrm{1}\right] \\ $$$$\mathrm{is}\:\mathrm{greater}\:\mathrm{than}\:\mathrm{2}\:\Rightarrow\:\mathrm{2}<\underset{−\mathrm{1}} {\int}^{\mathrm{1}} \sqrt{\mathrm{1}+{x}^{\mathrm{2}} }{dx}…
Question Number 129710 by SOMEDAVONG last updated on 18/Jan/21 $$\mathrm{I}=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{\mathrm{2304cosx}}{\left(\mathrm{cos4x}−\mathrm{8cos2x}+\mathrm{15}\right)^{\mathrm{2}} }\mathrm{dx}\: \\ $$ Answered by MJS_new last updated on 18/Jan/21 $$\mathrm{2304}\int\frac{\mathrm{cos}\:{x}}{\left(\mathrm{cos}\:\mathrm{4}{x}\:−\mathrm{8cos}\:\mathrm{2}{x}\:+\mathrm{15}\right)^{\mathrm{2}} }{dx}= \\…