Question Number 215051 by MrGaster last updated on 27/Dec/24 $$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\underset{{k}=\mathrm{1}} {\overset{{n}} {\prod}}\frac{{k}^{\mathrm{2}} +\mathrm{1}}{\:\sqrt{{k}^{\mathrm{2}} +\mathrm{4}}}=? \\ $$$$ \\ $$ Answered by MathematicalUser2357 last…
Question Number 215061 by mnjuly1970 last updated on 27/Dec/24 $$ \\ $$$$\int_{\mathrm{0}} ^{\:\infty} \int_{\mathrm{0}} ^{\:\infty} \mathrm{e}^{\:\:−\:\frac{\:{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} }{\mathrm{2}}} \mathrm{sin}\left({xy}\:\right){dxdy}=? \\ $$$$ \\ $$ Answered by…
Question Number 215072 by hardmath last updated on 27/Dec/24 $$\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{2000x}\:+\:\mathrm{1}\:=\:\mathrm{0} \\ $$$$\mathrm{Roots}:\:\:\boldsymbol{\mathrm{a}}\:\:\mathrm{and}\:\:\boldsymbol{\mathrm{b}} \\ $$$$\mathrm{x}^{\mathrm{2}} \:−\:\mathrm{2008x}\:−\:\mathrm{1}\:=\:\mathrm{0} \\ $$$$\mathrm{Roots}:\:\:\boldsymbol{\mathrm{c}}\:\:\mathrm{and}\:\:\boldsymbol{\mathrm{d}} \\ $$$$\mathrm{Find}:\:\:\left(\mathrm{a}+\mathrm{c}\right)\left(\mathrm{b}+\mathrm{d}\right)\left(\mathrm{a}−\mathrm{d}\right)\left(\mathrm{b}−\mathrm{c}\right)\:=\:? \\ $$ Commented by TonyCWX08…
Question Number 215032 by Tawa11 last updated on 26/Dec/24 Commented by Tawa11 last updated on 26/Dec/24 $$\mathrm{I}\:\mathrm{got}\:\:\mathrm{10}.\mathrm{3}\:\mathrm{m}/\mathrm{s} \\ $$ Answered by mr W last updated…
Question Number 215033 by MrGaster last updated on 27/Dec/24 $$ \\ $$$$\int_{\mathrm{0}} ^{\mathrm{2}\pi} \int_{\mathrm{0}} ^{\mathrm{2}\pi} \frac{\mathrm{sin}\theta−\mathrm{cos}\varphi−\mathrm{sin}\theta\mathrm{cos}\varphi}{\left(\mathrm{2sin}\theta−\mathrm{2cos}\varphi−\mathrm{2sin}\theta\mathrm{cos}\varphi+\mathrm{3}\right)^{\mathrm{3}/\mathrm{2}} }{d}\theta{d}\varphi \\ $$$$ \\ $$ Terms of Service Privacy…
Question Number 215020 by mnjuly1970 last updated on 26/Dec/24 $$ \\ $$$$\:\:\:\:{f}:\:\:\left[\mathrm{0}\:,\:\mathrm{1}\right]\:\rightarrow\mathbb{R}\:{is}\:{given}. \\ $$$$\:\:\:\:{f}\:''\:\:\:\:{is}\:{continuous}\:. \\ $$$$\:\:\:\:{by}\:{the}\:{way}\:\:{f}\left(\mathrm{0}\right)={f}\left(\mathrm{1}\right). \\ $$$$\:\:\:\:\:{prove}\:\:{that}\:: \\ $$$$\:\:\:\: \\ $$$$\begin{array}{|c|}{\int_{\mathrm{0}} ^{\:\mathrm{1}} \left(\:{f}\:''\:\left({x}\right)\right)^{\:\mathrm{2}} {dx}\:\geqslant\:\mathrm{3}\left({f}\:'\left(\mathrm{1}\right)\right)^{\mathrm{2}}…
Question Number 215041 by Red1ight last updated on 26/Dec/24 $$ \\ $$3 numbers are selected randomly from 0 to 10 (Continuous). forming a new range.…
Question Number 215004 by efronzo1 last updated on 25/Dec/24 Commented by mr W last updated on 26/Dec/24 $$\left({xy}+\frac{\lambda^{\mathrm{2}} }{{xy}}\right)_{{min}} =\frac{\mathrm{1}}{\mathrm{4}}+\mathrm{4}\lambda^{\mathrm{2}} \\ $$ Commented by efronzo1…
Question Number 215005 by Abdullahrussell last updated on 25/Dec/24 $$\:{a},{b},{c},{d}\in{R}\:{such}\:{that}, \\ $$$$\:\left({a}+{b}\right)\left({c}+{d}\right)=\mathrm{2} \\ $$$$\:\left({a}+{c}\right)\left({b}+{d}\right)=\mathrm{3} \\ $$$$\:\left({a}+{d}\right)\left({b}+{c}\right)=\mathrm{4}\: \\ $$$$\:{find}:\:\:\left({a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} +{d}^{\mathrm{2}} \right)_{{minimum}.} \\ $$ Answered…
Question Number 214997 by MrGaster last updated on 25/Dec/24 $$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\underset{{k}=−\infty} {\overset{+\infty} {\sum}}\frac{{k}^{\mathrm{2}} +{k}+\mathrm{1}}{{k}^{\mathrm{4}} +\mathrm{1}}=? \\ $$$$ \\ $$ Answered by mr W last…