Question Number 208431 by alcohol last updated on 15/Jun/24 $${h}_{{a}} \left({x}\right)\:=\:{e}^{−{x}} \:+\:{ax}^{\mathrm{2}} \\ $$$${show}\:{that}\:{h}_{{a}} \:{admits}\:{a}\:{minimum}\:{in}\:\mathbb{R} \\ $$ Answered by mathzup last updated on 15/Jun/24 $${cas}\:\mathrm{1}\:\:\:\:\:\:{a}>\mathrm{0}…
Question Number 208409 by hardmath last updated on 15/Jun/24 $$\mathrm{Find}:\:\:\:\int_{\mathrm{0}} ^{\:\mathrm{2}} \:\mid\mathrm{1}\:−\:\mathrm{x}\mid\:\mathrm{dx}\:=\:? \\ $$ Answered by mr W last updated on 15/Jun/24 $$=\mathrm{2}\int_{\mathrm{1}} ^{\mathrm{2}} \left({x}−\mathrm{1}\right){dx}…
Question Number 208420 by lepuissantcedricjunior last updated on 15/Jun/24 $$\boldsymbol{{resoudre}}\:\boldsymbol{{dans}}\:\mathbb{R}^{\mathrm{3}} \\ $$$$\begin{cases}{\boldsymbol{{x}}+\boldsymbol{{y}}=\mathrm{3}}\\{\boldsymbol{{y}}+\boldsymbol{{z}}=\mathrm{5}}\end{cases}\boldsymbol{{x}}+\boldsymbol{{z}}=\mathrm{4} \\ $$ Commented by A5T last updated on 15/Jun/24 $${You}\:{should}\:{learn}\:{to}\:{signify}\:{that}\:{you}\:{changed} \\ $$$${a}\:{question}. \\…
Question Number 208421 by lepuissantcedricjunior last updated on 15/Jun/24 Commented by Frix last updated on 15/Jun/24 $$\Psi=\mathrm{2}\underset{\mathrm{0}} {\overset{\frac{\pi}{\mathrm{4}}} {\int}}\frac{{dx}}{\:\sqrt{\mathrm{tan}\:\mathrm{2}{x}}}\:\overset{{t}=\sqrt{\mathrm{tan}\:\mathrm{2}{x}}} {=}\:\mathrm{2}\underset{\mathrm{0}} {\overset{\infty} {\int}}\frac{{dt}}{{t}^{\mathrm{4}} +\mathrm{1}}=\frac{\sqrt{\mathrm{2}}\pi}{\mathrm{2}} \\ $$…
Question Number 208423 by lepuissantcedricjunior last updated on 15/Jun/24 $$\:\:\:\boldsymbol{{calculons}}\: \\ $$$$\boldsymbol{{i}}=\int\int\int_{\left[\mathrm{0};\mathrm{1}\right]} \frac{\boldsymbol{{dxdydz}}}{\mathrm{1}−\boldsymbol{{xyz}}} \\ $$ Answered by Berbere last updated on 15/Jun/24 $$=\int\int\left[−\frac{\mathrm{1}}{{xy}}{ln}\left(\mathrm{1}−{xy}\right)\right]{dydx} \\ $$$${xy}={u}\Rightarrow{dy}=\frac{{du}}{{x}}…
Question Number 208418 by alcohol last updated on 16/Jun/24 $$\left.{u}_{{n}+\mathrm{1}} \:=\:{u}_{{n}} −{u}_{{n}} ^{\mathrm{3}} \:;\:{u}_{\mathrm{0}} \:\in\:\right]\mathrm{0},\:\mathrm{1}\left[\right. \\ $$$$\left..\:{show}\:{that}\:{u}_{{n}} \:\in\:\right]\mathrm{0},\:\mathrm{1}\left[\right. \\ $$$$.\:{show}\:{that}\:{u}_{{n}} \:{converges}\:{to}\:\mathrm{0} \\ $$$${v}_{{n}} \:=\:\frac{\mathrm{1}}{{u}_{{n}+\mathrm{1}} ^{\mathrm{2}}…
Question Number 208381 by hardmath last updated on 14/Jun/24 $$\mathrm{g}\left(\mathrm{x}\right)\:=\:\mathrm{lnx}^{\mathrm{2}} \\ $$$$\mathrm{f}\left(\mathrm{x}\right)\:=\:\sqrt[{\mathrm{3}}]{\mathrm{x}\:+\:\mathrm{25}} \\ $$$$\mathrm{Find}:\:\:\:\underset{\boldsymbol{\mathrm{x}}\rightarrow\boldsymbol{\mathrm{e}}} {\mathrm{lim}}\:\left(\mathrm{f}\left(\mathrm{g}\left(\mathrm{x}\right)\right)\:=\:?\right. \\ $$ Answered by A5T last updated on 14/Jun/24 $${f}\left({g}\left({x}\right)\right)=\sqrt[{\mathrm{3}}]{{ln}\left({x}^{\mathrm{2}}…
Question Number 208398 by mokys last updated on 14/Jun/24 $${write}\:{z}\:=\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{3}}+{i}}\:{in}\:{e}^{{i}\theta} \\ $$ Answered by A5T last updated on 14/Jun/24 $${z}=\frac{\sqrt{\mathrm{3}}−{i}}{\left(\sqrt{\mathrm{3}}\right)^{\mathrm{2}} −\left({i}\right)^{\mathrm{2}} }=\frac{\sqrt{\mathrm{3}}−{i}}{\mathrm{4}},\:\mid{z}\mid=\sqrt{\left(\frac{\sqrt{\mathrm{3}}}{\mathrm{4}}\right)^{\mathrm{2}} +\left(\frac{−\mathrm{1}}{\mathrm{4}}\right)^{\mathrm{2}} }=\frac{\mathrm{1}}{\mathrm{4}} \\…
Question Number 208377 by hardmath last updated on 14/Jun/24 $$\mathrm{sin}\:\mathrm{x}\:−\:\mathrm{sin}\:\frac{\pi}{\mathrm{6}}\:>\:\mathrm{0} \\ $$$$\mathrm{x}\:=\:? \\ $$ Commented by hardmath last updated on 14/Jun/24 $$ \\ $$Solve the…
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