Question Number 151448 by mathdanisur last updated on 21/Aug/21 $$\mathrm{Prove}\:\mathrm{that}: \\ $$$$\boldsymbol{\mathrm{A}}\mathrm{rtimetric}\:\mathrm{mean}\:\geqslant\:\boldsymbol{\mathrm{G}}\mathrm{eometric}\:\mathrm{mean} \\ $$$$\frac{\mathrm{a}\:+\:\mathrm{b}}{\mathrm{2}}\:\geqslant\:\sqrt{\mathrm{ab}} \\ $$ Commented by puissant last updated on 21/Aug/21 $$\forall\:\left({a},{b}\right)\in\mathbb{R}^{\mathrm{2}} ,…
Question Number 85915 by jagoll last updated on 26/Mar/20 $$\mathrm{3}^{\:^{\mid\mathrm{x}\mid} \mathrm{log}\:\mathrm{27}} \:\geqslant\:\frac{\mathrm{81}}{\mathrm{x}} \\ $$ Answered by john santu last updated on 26/Mar/20 Commented by jagoll…
Question Number 20375 by Tinkutara last updated on 26/Aug/17 $$\mathrm{A}\:\mathrm{small}\:\mathrm{particle}\:\mathrm{of}\:\mathrm{mass}\:{m}\:\mathrm{is}\:\mathrm{projected} \\ $$$$\mathrm{at}\:\mathrm{an}\:\mathrm{angle}\:\theta\:\mathrm{with}\:\mathrm{the}\:{x}-\mathrm{axis}\:\mathrm{with}\:\mathrm{an} \\ $$$$\mathrm{initial}\:\mathrm{velocity}\:{v}_{\mathrm{0}} \:\mathrm{in}\:\mathrm{the}\:{x}-{y}\:\mathrm{plane}\:\mathrm{as} \\ $$$$\mathrm{shown}\:\mathrm{in}\:\mathrm{the}\:\mathrm{Figure}.\:\mathrm{At}\:\mathrm{a}\:\mathrm{time} \\ $$$${t}\:<\:\frac{{v}_{\mathrm{0}} \:\mathrm{sin}\:\theta}{{g}},\:\mathrm{the}\:\mathrm{angular}\:\mathrm{momentum}\:\mathrm{of} \\ $$$$\mathrm{the}\:\mathrm{particle}\:\mathrm{is} \\ $$ Commented…
Question Number 151444 by liberty last updated on 21/Aug/21 Answered by mr W last updated on 21/Aug/21 $${say}\:{BE}={diameter}=\mathrm{2}×{AB} \\ $$$$\frac{{BC}}{{BO}}=\frac{{BO}}{{BE}} \\ $$$$\Rightarrow{BC}×{BE}={BO}^{\mathrm{2}} \\ $$$$\Rightarrow{BC}×\mathrm{2}×{AB}={BO}^{\mathrm{2}} \\…
Question Number 151447 by mnjuly1970 last updated on 21/Aug/21 $$ \\ $$$$\:\:\:\:\:\:{prove}\:{that}… \\ $$$$\:\:\:\mathrm{I}:=\:\int_{\mathrm{0}} ^{\:\infty} \frac{\:{ln}\:\left({x}\:\right)}{\mathrm{1}+\:{e}^{\:{x}} }\:{dx}=\:\frac{−\mathrm{1}}{\mathrm{2}}\:{ln}^{\:\mathrm{2}} \left(\mathrm{2}\right)\:..\blacksquare \\ $$ Answered by Lordose last updated…
Question Number 85909 by jagoll last updated on 26/Mar/20 $$\int\:\mathrm{sin}^{−\mathrm{1}} \:\left(\sqrt{\frac{\mathrm{x}}{\mathrm{a}+\mathrm{x}}}\right)\:\mathrm{dx}\:,\:\mathrm{a}\:>\:\mathrm{0} \\ $$ Commented by john santu last updated on 26/Mar/20 $${let}\:\sqrt{\frac{{x}}{{a}+{x}}}\:=\:{n}\:\Rightarrow\:{x}\:=\:\frac{{an}^{\mathrm{2}} }{\mathrm{1}−{n}^{\mathrm{2}} } \\…
Question Number 20372 by Tinkutara last updated on 26/Aug/17 $$\mathrm{The}\:\mathrm{two}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{an}\:\mathrm{equation}\:{x}^{\mathrm{3}} \:−\:\mathrm{9}{x}^{\mathrm{2}} \\ $$$$+\:\mathrm{14}{x}\:+\:\mathrm{24}\:=\:\mathrm{0}\:\mathrm{are}\:\mathrm{in}\:\mathrm{the}\:\mathrm{ratio}\:\mathrm{3}\::\:\mathrm{2}. \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{roots}. \\ $$ Answered by $@ty@m last updated on 26/Aug/17 $$\alpha+\mathrm{3}\beta+\mathrm{2}\beta=\mathrm{9}\:\Rightarrow\alpha+\mathrm{5}\beta=\mathrm{9}−−\left(\mathrm{1}\right)…
Question Number 20368 by Tinkutara last updated on 26/Aug/17 $$\mathrm{If}\:\alpha\:\mathrm{is}\:\mathrm{a}\:\mathrm{real}\:\mathrm{root}\:\mathrm{of}\:\mathrm{2}{x}^{\mathrm{3}} \:−\:\mathrm{3}{x}^{\mathrm{2}} \:+\:\mathrm{6}{x}\:+\:\mathrm{6}\:=\:\mathrm{0}, \\ $$$$\mathrm{then}\:\mathrm{find}\:\left[\alpha\right]\:\mathrm{where}\:\left[\centerdot\right]\:\mathrm{denotes}\:\mathrm{the} \\ $$$$\mathrm{greatest}\:\mathrm{integer}\:\mathrm{function}. \\ $$ Commented by ajfour last updated on 26/Aug/17…
Question Number 85902 by mr W last updated on 26/Mar/20 $${find}\:{the}\:{coefficients}\:{of}\:{x}^{\mathrm{2}} \:{and}\:{x}^{\mathrm{3}} \: \\ $$$${terms}\:{in}\:{the}\:{expansion}\:{of} \\ $$$$\left(\mathrm{1}+{x}\right)\left(\mathrm{1}+\mathrm{2}{x}\right)^{\mathrm{2}} \left(\mathrm{1}+\mathrm{3}{x}\right)^{\mathrm{3}} …\left(\mathrm{1}+\mathrm{100}{x}\right)^{\mathrm{100}} \\ $$ Commented by Serlea last…
Question Number 20367 by Tinkutara last updated on 26/Aug/17 $$\mathrm{Let}\:{f}\left({x}\right)\:=\:{x}^{\mathrm{3}} \:+\:\mathrm{3}{x}^{\mathrm{2}} \:+\:\mathrm{9}{x}\:+\:\mathrm{6sin}{x},\:\mathrm{then} \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{real}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{equation} \\ $$$$\frac{\mathrm{1}}{{x}\:−\:{f}\left(\mathrm{1}\right)}\:+\:\frac{\mathrm{2}}{{x}\:−\:{f}\left(\mathrm{2}\right)}\:+\:\frac{\mathrm{3}}{{x}\:−\:{f}\left(\mathrm{3}\right)}\:=\:\mathrm{0}. \\ $$ Answered by ajfour last updated…