Question Number 49389 by behi83417@gmail.com last updated on 06/Dec/18 $${for}\:{x}\neq\mathrm{0},{y}\neq\mathrm{0},\mathrm{xy}\neq−\mathrm{1},{f}\left(\mathrm{1}\right)=\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\boldsymbol{\mathrm{f}}\left(\boldsymbol{\mathrm{x}}\right)+\boldsymbol{\mathrm{f}}\left(\boldsymbol{\mathrm{y}}\right)=\boldsymbol{\mathrm{f}}\left(\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{y}}\right)+\frac{\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{y}}}{\mathrm{1}+\boldsymbol{\mathrm{xy}}} \\ $$$$\mathrm{1}.{find}:\:{f}\left({x}\right),\left[{if}\:{possible}\right] \\ $$$$\mathrm{2}.{find}\::{f}^{−\mathrm{1}} \left(\mathrm{1}\right),\left[{if}\:{possible}\right]. \\ $$ Answered by kaivan.ahmadi last updated on…
Question Number 114878 by bobhans last updated on 21/Sep/20 $${Given}\:{f}\left({x}\right)\:=\:\underset{\mathrm{0}} {\overset{{x}} {\int}}\:\frac{{dt}}{\:\sqrt{\mathrm{1}+{t}^{\mathrm{3}} }}\:{and}\:{g}\left({x}\right)\:{be}\:{the} \\ $$$${inverse}\:{function}\:{of}\:{f}\left({x}\right),\:{then}\:{g}\:''\left({x}\right)=\lambda{g}^{\mathrm{2}} \left({x}\right). \\ $$$${then}\:{the}\:{value}\:{of}\:\lambda\:= \\ $$ Answered by Olaf last updated…
Question Number 49095 by maxmathsup by imad last updated on 02/Dec/18 $${prove}\:{the}\:{existence}\:{of}\:{n}\:{integrs}\:{naturals}\:{x}_{\mathrm{1}} ,{x}_{\mathrm{2}} ,….{x}_{{n}} \:\:\:\:{with}\:{x}_{{i}} \neq{x}_{{j}} {for}\:{i}\neq{j} \\ $$$${and}\:\frac{\mathrm{1}}{{x}_{\mathrm{1}} }\:+\frac{\mathrm{1}}{{x}_{\mathrm{2}} }\:+….+\frac{\mathrm{1}}{{x}_{{n}} }\:=\mathrm{1}\:. \\ $$ Answered…
Question Number 114597 by O Predador last updated on 19/Sep/20 $$\: \\ $$$$\:\:\:\boldsymbol{\mathrm{What}}\:\:\boldsymbol{\mathrm{is}}\:\:\boldsymbol{\mathrm{the}}\:\:\boldsymbol{\mathrm{number}}\:\:\boldsymbol{\mathrm{value}}\:\:\boldsymbol{\mathrm{of}}\:\:\:\boldsymbol{\mathrm{f}}\left[\sqrt[{\mathrm{3}}]{\boldsymbol{\mathrm{log}}\left(\frac{\:\sqrt{\boldsymbol{\mathrm{x}}\:\:}\:−\:\:\:\mathrm{1}\:}{\boldsymbol{\mathrm{x}}\:\:\:−\:\:\:\mathrm{1}}\right)\:\:}\right]\:\:=\:\:\sqrt{\sqrt{\boldsymbol{\mathrm{x}}\:}\:\:\:+\:\:\:\boldsymbol{\mathrm{x}}\:}\:\:\boldsymbol{\mathrm{for}}\:\:\boldsymbol{\mathrm{f}}\left(−\mathrm{1}\right)? \\ $$$$\: \\ $$$$\left.\:\:\:\boldsymbol{\mathrm{a}}\right)\:\mathrm{0},\mathrm{1} \\ $$$$\left.\:\:\:\boldsymbol{\mathrm{b}}\right)\:\mathrm{27} \\ $$$$\left.\:\:\:\boldsymbol{\mathrm{c}}\right)\:\mathrm{81} \\ $$$$\left.\:\:\:\boldsymbol{\mathrm{d}}\right)\:\mathrm{10} \\ $$$$\left.\:\:\:\boldsymbol{\mathrm{e}}\right)\:\mathrm{12}…
Question Number 48668 by maxmathsup by imad last updated on 26/Nov/18 $${let}\:{f}\left({x}\right)=\frac{{e}^{−\mathrm{2}{x}} }{{x}+\mathrm{1}} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}^{\left({n}\right)} \left({x}\right)\:{and}\:{f}^{\left({n}\right)} \left(\mathrm{0}\right)\:. \\ $$$$\left.\mathrm{2}\right)\:{develop}\:{f}\:{at}\:{integr}\:{serie}\:. \\ $$ Commented by maxmathsup by…
Question Number 48506 by Abdo msup. last updated on 24/Nov/18 $${let}\:\:{S}_{{n}} =\sum_{{k}=\mathrm{0}} ^{\infty} \:\frac{\left(−\mathrm{1}\right)^{{k}} }{\mathrm{2}{k}+\mathrm{1}} \\ $$$$\left.\mathrm{1}\right){prove}\:{that}\:\frac{\pi}{\mathrm{4}}\:−{S}_{{n}} =\left(−\mathrm{1}\right)^{{n}+\mathrm{1}} \:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{t}^{\mathrm{2}{n}+\mathrm{2}} }{\mathrm{1}+{t}^{\mathrm{2}} }{dt} \\ $$$$\left.\mathrm{2}\right)\:{conclude}\:{lim}_{{n}\rightarrow+\infty}…
Question Number 48493 by maxmathsup by imad last updated on 24/Nov/18 $${let}\:{S}_{{n}} =\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\frac{{k}^{\mathrm{2}} }{\left(\mathrm{2}{k}−\mathrm{1}\right)\left(\mathrm{2}{k}+\mathrm{1}\right)} \\ $$$$\left.\mathrm{1}\right)\:{determine}\:{S}_{{n}} \:{interms}\:{of}\:{n} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{lim}_{{n}\rightarrow+\infty} \:{S}_{{n}} \\ $$ Commented…
Question Number 113904 by bobhans last updated on 16/Sep/20 $${If}\:\begin{cases}{{f}\left({x}\right)=\sqrt{\mathrm{2}{x}−\mathrm{5}}}\\{{g}\left({x}\right)={x}^{\mathrm{2}} +\mathrm{1}}\end{cases} \\ $$$${find}\:\left({f}^{−\mathrm{1}} \circ{g}\right)^{−\mathrm{1}} \left({x}\right) \\ $$ Answered by bemath last updated on 16/Sep/20 $${solution}\::\:…
Question Number 48267 by maxmathsup by imad last updated on 21/Nov/18 $${f}\:{is}\:{a}\:{function}\:{verify}\:{f}\left({x}+\mathrm{1}\right)\:+{x}^{\mathrm{2}} =\mathrm{3}{f}\left({x}\right) \\ $$$$\left.\mathrm{1}\right){find}\:{f}\left(\mathrm{8}\right)\:{and}\:{f}\left(\mathrm{12}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\sum_{{k}=\mathrm{0}} ^{{n}} {f}\left({k}\right) \\ $$$$\left.\mathrm{3}\right)\:{find}\:\sum_{{k}=\mathrm{0}} ^{{n}} \:{f}^{\mathrm{2}} \left({k}\right)\:. \\…
Question Number 48174 by Abdo msup. last updated on 20/Nov/18 $${find}\:{lim}_{{n}\rightarrow+\infty} \:\:\:\int_{\mathrm{0}} ^{{n}} \:\:\:{sin}\left(\frac{\pi{x}}{{n}}\right){dx}\:. \\ $$ Commented by Abdo msup. last updated on 20/Nov/18 $${let}\:{A}_{{n}}…