Question Number 151767 by mathdanisur last updated on 22/Aug/21 $$\underset{\boldsymbol{\mathrm{n}}\rightarrow\infty} {\mathrm{lim}}\:\underset{\boldsymbol{\mathrm{k}}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{\mathrm{n}}{\mathrm{n}^{\mathrm{2}} \:+\:\mathrm{k}}\:=\:? \\ $$ Answered by Olaf_Thorendsen last updated on 22/Aug/21 $$\mathrm{my}\:\mathrm{calculous}\:\mathrm{was}\:\mathrm{false}. \\…
Question Number 151766 by mathdanisur last updated on 22/Aug/21 Commented by tabata last updated on 23/Aug/21 $${ln}\left({y}\right)\:{ln}\left({x}\right)={ln}\left(\mathrm{3}\right)\Rightarrow{ln}\left({y}\right)=\frac{{ln}\left(\mathrm{3}\right)}{{ln}\left({x}\right)} \\ $$$$ \\ $$$$\Rightarrow\frac{{y}^{'} }{{y}}=−\frac{{ln}\left(\mathrm{3}\right)}{{x}\:\left({ln}\left({x}\right)\right)^{\mathrm{2}} } \\ $$$$…
Question Number 151758 by mathdanisur last updated on 22/Aug/21 $$\mathrm{f}\left(\mathrm{3x}+\mathrm{1}\right)=\mathrm{g}^{−\mathrm{1}} \left(\mathrm{5x}^{\mathrm{2}} −\mathrm{2}\right) \\ $$$$\left({g}\:{o}\:{f}\right)^{'} \:\left(\mathrm{4}\right)\:=\:? \\ $$ Commented by otchereabdullai@gmail.com last updated on 23/Aug/21 $$\mathrm{nice}!…
Question Number 20684 by Tinkutara last updated on 31/Aug/17 $${If}\:{the}\:{equation}\:{x}^{\mathrm{2}} \:+\:\beta^{\mathrm{2}} \:=\:\mathrm{1}\:−\:\mathrm{2}\beta{x}\:{and} \\ $$$${x}^{\mathrm{2}} \:+\:\alpha^{\mathrm{2}} \:=\:\mathrm{1}\:−\:\mathrm{2}\alpha{x}\:{have}\:{one}\:{and}\:{only} \\ $$$${one}\:{root}\:{in}\:{common},\:{then}\:\mid\alpha\:−\:\beta\mid\:{is} \\ $$$${equal}\:{to} \\ $$ Answered by $@ty@m…
Question Number 20671 by Tinkutara last updated on 31/Aug/17 $${The}\:{total}\:{number}\:{of}\:{positive}\:{integral} \\ $$$${solution}\left({s}\right)\:{of}\:{the}\:{inequation} \\ $$$$\frac{{x}^{\mathrm{2}} \left(\mathrm{3}{x}\:−\:\mathrm{4}\right)^{\mathrm{3}} \left({x}\:−\:\mathrm{2}\right)^{\mathrm{4}} }{\left({x}\:−\:\mathrm{5}\right)^{\mathrm{5}} \left(\mathrm{2}{x}\:−\:\mathrm{7}\right)^{\mathrm{6}} }\:\leqslant\:\mathrm{0}\:{is}/{are} \\ $$ Answered by ajfour last…
Question Number 86206 by behi83417@gmail.com last updated on 27/Mar/20 $$\mathrm{1}.\mathrm{line}:\boldsymbol{\mathrm{y}}=−\boldsymbol{\mathrm{x}}+\mathrm{4}\:\:,\mathrm{meets}\::\:\boldsymbol{\mathrm{xy}}=\mathrm{1}\:\mathrm{at}:\boldsymbol{\mathrm{A}},\boldsymbol{\mathrm{B}}. \\ $$$$\:\:\:\:\:\:\Rightarrow\:\:\mathrm{S}_{\mathrm{O}\overset{\bigtriangleup} {\mathrm{A}B}} =?\:\left(\mathrm{O}=\mathrm{origin}\:\mathrm{of}\:\mathrm{cordinates}\right) \\ $$$$\mathrm{2}.\mathrm{find}\::\mathrm{center}\:\mathrm{area}\:\mathrm{of}\:\mathrm{region}\:\mathrm{bonded}\:\mathrm{by} \\ $$$$\mathrm{corve}:\:\:\sqrt{\frac{\boldsymbol{\mathrm{x}}}{\boldsymbol{\mathrm{a}}}}+\sqrt{\frac{\boldsymbol{\mathrm{y}}}{\boldsymbol{\mathrm{b}}}}=\mathrm{1},\mathrm{and}\:\boldsymbol{\mathrm{x}},\boldsymbol{\mathrm{y}}\:\mathrm{axes}. \\ $$$$\left(\boldsymbol{\mathrm{a}}\neq\boldsymbol{\mathrm{b}}\right)\in\boldsymbol{\mathrm{R}}^{+} \\ $$ Commented by jagoll…
Question Number 151747 by mathdanisur last updated on 22/Aug/21 $$\mathrm{let}\:\:\mathrm{f}\left(\mathrm{x}\right)=\frac{\lambda-\mathrm{x}}{\mathrm{1}+\mathrm{x}^{\mathrm{2}} }\:\:\mathrm{and}\:\:\lambda\geqslant\frac{-\mathrm{3}}{\mathrm{4}} \\ $$$$\mathrm{solve}\:\mathrm{in}\:\mathbb{R}\:\:\mathrm{f}\left(\mathrm{f}\left(\mathrm{f}\left(\mathrm{x}\right)\right)\right)\:\leqslant\:\mathrm{0} \\ $$ Answered by mr W last updated on 22/Aug/21 $${say}\:{g}\left({x}\right)={f}\left({f}\left({x}\right)\right) \\…
Question Number 20670 by Tinkutara last updated on 31/Aug/17 $${For}\:{the}\:{equation}\:\mathrm{3}{x}^{\mathrm{2}} \:+\:{px}\:+\:\mathrm{3}\:=\:\mathrm{0}, \\ $$$${find}\:{the}\:{value}\left({s}\right)\:{of}\:{p}\:{if}\:{one}\:{root}\:{is} \\ $$$$\left({i}\right)\:{square}\:{of}\:{the}\:{other} \\ $$$$\left({ii}\right)\:{fourth}\:{power}\:{of}\:{the}\:{other}. \\ $$ Answered by dioph last updated on…
Question Number 20652 by Tinkutara last updated on 30/Aug/17 $$\mathrm{Suppose}\:{x}\:\mathrm{is}\:\mathrm{a}\:\mathrm{positive}\:\mathrm{real}\:\mathrm{number} \\ $$$$\mathrm{such}\:\mathrm{that}\:\left\{{x}\right\},\:\left[{x}\right]\:\mathrm{and}\:{x}\:\mathrm{are}\:\mathrm{in}\:\mathrm{a} \\ $$$$\mathrm{geometric}\:\mathrm{progression}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{least} \\ $$$$\mathrm{positive}\:\mathrm{integer}\:{n}\:\mathrm{such}\:\mathrm{that}\:{x}^{{n}} \:>\:\mathrm{100}. \\ $$$$\left(\mathrm{Here}\:\left[{x}\right]\:\mathrm{denotes}\:\mathrm{the}\:\mathrm{integer}\:\mathrm{part}\:\mathrm{of}\:{x}\right. \\ $$$$\left.\mathrm{and}\:\left\{{x}\right\}\:=\:{x}\:−\:\left[{x}\right].\right) \\ $$ Commented by…
Question Number 20631 by Tinkutara last updated on 30/Aug/17 $${Solve}\:{the}\:{inequality} \\ $$$$\left({x}\:+\:\mathrm{3}\right)^{\mathrm{5}} \:−\:\left({x}\:−\:\mathrm{1}\right)^{\mathrm{5}} \:\geqslant\:\mathrm{244}. \\ $$ Answered by mrW1 last updated on 30/Aug/17 $$\mathrm{let}\:\mathrm{u}=\mathrm{x}+\mathrm{1} \\…