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Question Number 119848 by benjo_mathlover last updated on 27/Oct/20 $${Solve}\:{in}\:{real}\:{numbers}\:{the}\:{equation} \\ $$$$\sqrt[{\mathrm{3}\:}]{{x}}\:+\:\sqrt[{\mathrm{3}\:}]{{x}−\mathrm{1}}\:+\:\sqrt[{\mathrm{3}\:}]{{x}+\mathrm{1}}\:=\:\mathrm{0} \\ $$ Answered by 1549442205PVT last updated on 27/Oct/20 $$\mathrm{Applying}\:\mathrm{the}\:\mathrm{identity}\: \\ $$$$\mathrm{a}^{\mathrm{3}} +\mathrm{b}^{\mathrm{3}}…
Question Number 119849 by benjo_mathlover last updated on 27/Oct/20 $${Find}\:{all}\:{pair}\left({x},{y}\right)\:{of}\:{real}\:{numbers} \\ $$$${that}\:{are}\:{the}\:{solutions}\:{to}\:{the}\:{system} \\ $$$$\begin{cases}{{x}^{\mathrm{4}} +\mathrm{2}{x}^{\mathrm{3}} −{y}=−\frac{\mathrm{1}}{\mathrm{4}}+\sqrt{\mathrm{3}}}\\{{y}^{\mathrm{4}} +\mathrm{2}{y}^{\mathrm{3}} −{x}=−\frac{\mathrm{1}}{\mathrm{4}}−\sqrt{\mathrm{3}}}\end{cases} \\ $$ Answered by 1549442205PVT last updated…
Question Number 185383 by aba last updated on 20/Jan/23 $$\underset{\mathrm{n}\rightarrow+\infty} {\mathrm{lim}}\frac{\sqrt[{\mathrm{n}}]{\left(_{\mathrm{1}} ^{\mathrm{n}} \right)\left(_{\mathrm{2}} ^{\mathrm{n}} \right)…\left(_{\mathrm{n}} ^{\mathrm{n}} \right)}}{\mathrm{e}^{\frac{\mathrm{n}}{\mathrm{2}}} ×\mathrm{n}^{−\frac{\mathrm{1}}{\mathrm{2}}} }=? \\ $$ Terms of Service Privacy…
Question Number 119832 by talminator2856791 last updated on 27/Oct/20 $$\: \\ $$$$\: \\ $$$$\:\:\:\:\:\:\:\:\mathrm{evaluate}:\:\:\:\:{I}\:\:=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \left(\frac{{x}+\mathrm{1}}{{x}}\right)^{{x}!} {dx} \\ $$$$ \\ $$$$ \\ $$ Terms of…
Question Number 119831 by talminator2856791 last updated on 27/Oct/20 $$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\mathrm{evaluate}:\:\:\:{I}\:\:=\:\int_{\mathrm{1}} ^{\:\infty} \left(\frac{\mathrm{1}}{{x}}\right)^{{x}} {dx} \\ $$$$\: \\ $$$$\: \\ $$…
Question Number 54278 by ajfour last updated on 01/Feb/19 $${If}\:{f}\:{be}\:{n}\rightarrow{n}\:\:\:\left({n}\in\mathbb{N}\right)\:{and}\:{is}\:{increasing}, \\ $$$${f}\left({f}\left({n}\right)\right)=\mathrm{3}{n};\:{find}\:{f}\left(\mathrm{2}\right). \\ $$$${options}:\:\:\mathrm{1},\:\mathrm{2},\:\mathrm{3},\mathrm{4}\:. \\ $$ Commented by mr W last updated on 02/Feb/19 $${if}\:{f}\left({f}\left({x}\right)\right)=\mathrm{3}{x}…
Question Number 119807 by Ar Brandon last updated on 27/Oct/20 $$\mathrm{Let}\:{f}\:\mathrm{be}\:\mathrm{a}\:\mathrm{real}-\mathrm{valued}\:\mathrm{function}\:\mathrm{defined}\:\mathrm{on}\:\mathrm{the}\:\mathrm{inte}- \\ $$$$\mathrm{rval}\:\left[−\mathrm{1},\:\mathrm{1}\right].\:\mathrm{If}\:\mathrm{the}\:\mathrm{area}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equilateral}\:\mathrm{triangle}\:\mathrm{with} \\ $$$$\left(\mathrm{0},\:\mathrm{0}\right)\:\mathrm{and}\:\left(\mathrm{x},\:{f}\left(\mathrm{x}\right)\right)\:\mathrm{as}\:\mathrm{two}\:\mathrm{vertices}\:\mathrm{is}\:\sqrt{\mathrm{3}}/\mathrm{4},\:\mathrm{then}\:{f}\left(\mathrm{x}\right) \\ $$$$\mathrm{is}\:\mathrm{equal}\:\mathrm{to} \\ $$$$\left(\mathrm{A}\right)\:\sqrt{\mathrm{1}−\mathrm{x}^{\mathrm{2}} }\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{B}\right)\:\sqrt{\mathrm{1}+\mathrm{x}^{\mathrm{2}} } \\ $$$$\left(\mathrm{C}\right)\:−\sqrt{\mathrm{1}−\mathrm{x}^{\mathrm{2}} }\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{D}\right)\:−\sqrt{\mathrm{1}+\mathrm{x}^{\mathrm{2}} }…
Question Number 54271 by naveen200601 last updated on 01/Feb/19 Commented by Meritguide1234 last updated on 02/Feb/19 $$\mathrm{post}\:\mathrm{many}\:\mathrm{problem}\:\mathrm{on}\:\mathrm{this}\:\mathrm{type} \\ $$ Answered by tanmay.chaudhury50@gmail.com last updated on…
Question Number 185337 by Shrinava last updated on 20/Jan/23 Answered by qaz last updated on 20/Jan/23 $$\underset{{k}=\mathrm{2}} {\overset{{n}} {\sum}}\frac{{k}+\sqrt{{k}}}{{k}}\sim\int_{\mathrm{2}} ^{{n}} \frac{{x}+\sqrt{{x}}}{{x}}{dx}\sim{n} \\ $$$$\underset{{k}=\mathrm{2}} {\overset{{n}} {\sum}}\frac{{k}+\sqrt[{\mathrm{3}}]{{k}}}{{k}}\sim\int_{\mathrm{2}}…