Question Number 130593 by mr W last updated on 27/Jan/21 $${if}\:{a}_{\mathrm{0}} =\mathrm{1},\:{a}_{\mathrm{1}} =\mathrm{2}\:{and}\:{a}_{{n}+\mathrm{1}} =\sqrt{{a}_{{n}} {a}_{{n}−\mathrm{1}} } \\ $$$${find}\:{a}_{{n}} \:{in}\:{terms}\:{of}\:{n}. \\ $$ Commented by malwan last…
Question Number 130496 by Adel last updated on 26/Jan/21 $$\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{x}!\right)=? \\ $$ Answered by MJS_new last updated on 26/Jan/21 $${x}!\:\mathrm{is}\:\mathrm{defined}\:\mathrm{for}\:{x}\in\mathbb{N}\:\Rightarrow\:\mathrm{no}\:\mathrm{derivate}\:\mathrm{exists} \\ $$$$ \\ $$$$\mathrm{if}\:\mathrm{you}\:\mathrm{mean}\:{x}!=\Gamma\:\left({x}+\mathrm{1}\right) \\…
Question Number 130476 by shaker last updated on 26/Jan/21 Answered by TheSupreme last updated on 26/Jan/21 $$\left(\frac{\mathrm{3}}{\mathrm{5}}\right)^{{x}} +\left(\frac{\mathrm{4}}{\mathrm{5}}\right)^{{x}} =\mathrm{1} \\ $$$${f}\left({x}\right)=\left(\frac{\mathrm{3}}{\mathrm{5}}\right)^{{x}} +\left(\frac{\mathrm{4}}{\mathrm{5}}\right)^{{x}} \\ $$$${f}'\left({x}\right)<\mathrm{0}\:\forall{x}\in\mathbb{R} \\…
Question Number 130472 by shaker last updated on 26/Jan/21 Answered by mathmax by abdo last updated on 26/Jan/21 $$\mathrm{let}\:\mathrm{w}\left(\mathrm{x}\right)\:=\sum_{\mathrm{k}=\mathrm{1}} ^{\mathrm{n}} \left(\mathrm{2k}−\mathrm{1}\right)\mathrm{x}^{\mathrm{k}} \:\Rightarrow\mathrm{w}\left(\mathrm{x}\right)=\mathrm{2}\sum_{\mathrm{k}=\mathrm{1}} ^{\mathrm{n}} \:\mathrm{kx}^{\mathrm{k}} −\sum_{\mathrm{k}=\mathrm{1}}…
Question Number 130455 by ayoubbacmath0 last updated on 25/Jan/21 $$\mathrm{f}\left(\mathrm{x}\right)=\mathrm{x}+\mathrm{ln}\mid\mathrm{e}^{\mathrm{x}} −\mathrm{2}\mid \\ $$$$\begin{cases}{\left.\mathrm{f}\left(\mathrm{x}\right)=\mathrm{x}+\mathrm{ln}\left(\mathrm{e}^{\mathrm{x}} −\mathrm{2}\right)\:\:\:\:\:\:\mathrm{x}\in\right]\mathrm{ln2};+\infty\left[\right.}\\{\left.\mathrm{f}\left(\mathrm{x}\right)=\mathrm{x}+\mathrm{ln}\left(−\mathrm{e}^{\mathrm{x}} +\mathrm{2}\right)\:\:\:\mathrm{x}\in\right]−\infty;\mathrm{ln2}\left[\right.}\end{cases} \\ $$$$\underset{{x}\rightarrow+\infty} {\mathrm{lim}f}\left(\mathrm{x}\right)=\underset{{x}\rightarrow+\infty} {\mathrm{lim}}\left(\mathrm{x}+\mathrm{ln}\left(\mathrm{e}^{\mathrm{x}} −\mathrm{2}\right)\right)=+\infty \\ $$$$\underset{{x}\rightarrow−\infty} {\mathrm{lim}f}\left(\mathrm{x}\right)=\underset{{x}\rightarrow−\infty} {\mathrm{lim}}\left(\mathrm{x}+\mathrm{ln}\left(−\mathrm{e}^{\mathrm{x}} +\mathrm{2}\right)\right)=−\infty…
Question Number 130442 by greg_ed last updated on 25/Jan/21 $${g}\left({x}\right)=\mathrm{3}{x}−\mathrm{2}. \\ $$$$\mathrm{determine}\:{a}\:\mathrm{et}\:{b}\:\mathrm{such}\:\mathrm{that}\:{a}\:\leqslant\:{g}\left({x}\right)\:\leqslant\:{b}. \\ $$ Answered by mr W last updated on 25/Jan/21 $$\underset{{x}\rightarrow−\infty} {\mathrm{lim}}{g}\left({x}\right)=−\infty \\…
Question Number 130426 by Koyoooo last updated on 25/Jan/21 Answered by Olaf last updated on 25/Jan/21 $$\left(\mathrm{9}×\mathrm{9}×\mathrm{9}×\mathrm{9}\right)_{\mathrm{10}} \:=\:\mathrm{100}_{\mathrm{81}} \\ $$$$\mathrm{9}^{\mathrm{4}} \:\mathrm{en}\:\mathrm{base}\:\mathrm{10}\:\mathrm{egale}\:\mathrm{100}\:\mathrm{en}\:\mathrm{base}\:\mathrm{81} \\ $$ Answered by…
Question Number 64867 by meme last updated on 22/Jul/19 $${x}^{\mathrm{4}} +\left(\mathrm{2}{i}−\mathrm{3}\right){x}^{\mathrm{3}} −\left(\mathrm{1}+\mathrm{6}{i}\right){x}^{\mathrm{2}} +\left(\mathrm{3}−\mathrm{2}{i}\right){x}−\mathrm{2}=\mathrm{0} \\ $$ Answered by MJS last updated on 22/Jul/19 $$\mathrm{no}\:“\mathrm{nice}''\:\mathrm{exact}\:\mathrm{solution} \\ $$$${x}_{\mathrm{1}}…
Question Number 130383 by MrHusseinElmasry last updated on 25/Jan/21 $${If}\:\left({a}−\mathrm{2}\right)+\mathrm{3}{i}=\mathrm{5}−{bi}\:{then}\:{a}+{b}= \\ $$ Answered by mathmax by abdo last updated on 25/Jan/21 $$\mathrm{a}−\mathrm{2}+\mathrm{3i}=\mathrm{5}−\mathrm{bi}\:\Rightarrow\mathrm{a}−\mathrm{2}+\mathrm{3i}−\mathrm{5}+\mathrm{bi}=\mathrm{0}\:\Rightarrow\mathrm{a}−\mathrm{7}+\mathrm{i}\left(\mathrm{3}+\mathrm{b}\right)=\mathrm{0}\:\Rightarrow \\ $$$$\begin{cases}{\mathrm{a}−\mathrm{7}=\mathrm{0}}\\{\mathrm{3}+\mathrm{b}=\mathrm{0}\:\:\:\Rightarrow\begin{cases}{\mathrm{a}=\mathrm{7}}\\{\mathrm{b}=−\mathrm{3}\:}\end{cases}}\end{cases} \\…
Question Number 64837 by aliesam last updated on 22/Jul/19 $$\begin{cases}{{x}^{\sqrt{{y}}} \:+\:{y}^{\sqrt{{x}}} \:=\:\frac{\mathrm{49}}{\mathrm{48}}}\\{\sqrt{{x}}\:+\:\sqrt{{y}}\:=\:\frac{\mathrm{7}}{\mathrm{2}}}\end{cases} \\ $$$$ \\ $$$${find}\:{x}\:{and}\:{y} \\ $$$$ \\ $$ Answered by MJS last updated…