Question Number 144634 by loveineq last updated on 27/Jun/21 $$\mathrm{Let}\:{a},{b},{c}\:>\:\mathrm{0}\:\mathrm{and}\:\left({a}+{b}\right)\left({b}+{c}\right)\:=\:\mathrm{4}.\:\mathrm{Prove}\:\mathrm{that}\:\:\:\:\:\:\:\: \\ $$$$\left(\mathrm{1}\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{a}^{\mathrm{2}} +\mathrm{2}{b}^{\mathrm{2}} +{c}^{\mathrm{2}} +\frac{\mathrm{2}{b}\left({c}^{\mathrm{2}} +{a}^{\mathrm{2}} \right)}{{c}+{a}}\:\geqslant\:\mathrm{6} \\ $$$$\left(\mathrm{2}\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{a}^{\mathrm{3}} +\mathrm{3}{b}^{\mathrm{3}} +{c}^{\mathrm{3}} +\frac{\mathrm{3}{b}\left({c}^{\mathrm{3}} +{a}^{\mathrm{3}} \right)}{{c}+{a}}\:\geqslant\:\mathrm{8} \\…
Question Number 144629 by mathdanisur last updated on 27/Jun/21 $${if}\:\:{f}^{\mathrm{2}} \left(\mathrm{2}{x}-\mathrm{1}\right)-\mathrm{10}{f}\left(\mathrm{3}{x}-\mathrm{2}\right)+\mathrm{25}=\mathrm{0} \\ $$$${find}\:\:{f}\:^{'} \left(\mathrm{1}\right)+{f}\left(\mathrm{1}\right)=? \\ $$ Answered by liberty last updated on 27/Jun/21 $$\left[\mathrm{f}\left(\mathrm{2x}−\mathrm{1}\right)\right]^{\mathrm{2}} −\mathrm{10f}\left(\mathrm{3x}−\mathrm{2}\right)+\mathrm{25}=\mathrm{0}…
Question Number 79089 by TawaTawa last updated on 22/Jan/20 Commented by mr W last updated on 22/Jan/20 $${set}\:\mathrm{2}: \\ $$$${standard}\:{deviation}\:\sigma_{\mathrm{2}} =\mathrm{2}\sigma_{\mathrm{1}} =\mathrm{2}×\mathrm{3}.\mathrm{5}=\mathrm{7} \\ $$$${variance}\:\sigma_{\mathrm{2}} ^{\mathrm{2}}…
Question Number 79085 by key of knowledge last updated on 22/Jan/20 $$\alpha\:\mathrm{and}\:\beta\:\left(\mathrm{or}\:\mathrm{more}\right)\:\:\mathrm{are}\:\mathrm{root}\:\mathrm{of}: \\ $$$$\left(\mathrm{x}^{\mathrm{2}} +\mathrm{x}\right)^{\mathrm{2}} −\mathrm{2}\left(\mathrm{x}^{\mathrm{2}} +\mathrm{x}\right)−\mathrm{k}=\mathrm{0} \\ $$$$\mathrm{and}\:\:\alpha\beta<\mathrm{0}\:\:\:\:\:\:\:\:\Rightarrow\mathrm{k}\in? \\ $$ Commented by mr W…
Question Number 144609 by mathdanisur last updated on 26/Jun/21 $${How}\:{many}\:{digits}\:{doest}\:{the}\:{number} \\ $$$$\mathrm{2021}^{\mathrm{2022}} \:\:{have}.? \\ $$ Answered by MJS_new last updated on 27/Jun/21 $$\mathrm{log}_{\mathrm{10}} \:\mathrm{2021}^{\mathrm{2022}} \:=\mathrm{2022log}_{\mathrm{10}}…
Question Number 144600 by mathdanisur last updated on 26/Jun/21 $${if}\:{x},{y},{z}>\mathrm{0}\:;\:{xy}+{yz}+{zx}=\mathrm{1}\:{prove}\:{that}: \\ $$$${xyz}\:+\:\sqrt[{\mathrm{3}}]{\left(\mathrm{1}+{x}^{\mathrm{3}} \right)\left(\mathrm{1}+{y}^{\mathrm{3}} \right)\left(\mathrm{1}+{z}^{\mathrm{3}} \right)}\:\geqslant\:\mathrm{1} \\ $$ Answered by mindispower last updated on 27/Jun/21 $${x},{y},{z}>\mathrm{0}…
Question Number 144603 by loveineq last updated on 26/Jun/21 $$\mathrm{Let}\:{a},{b},{c}\:>\:\mathrm{0}\:\mathrm{and}\:\left({a}+{b}\right)\left({b}+{c}\right)\:=\:\mathrm{4}.\:\mathrm{Prove}\:\mathrm{that} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{2}{a}+{b}\right)\left({a}+{b}\right)+\left({b}+\mathrm{2}{c}\right)\left({b}+{c}\right)\:\geqslant\:\mathrm{8}+\frac{\mathrm{1}}{\mathrm{2}}\left({a}+\mathrm{2}{b}+{c}\right)\left({c}+{a}\right)\:\:\:\:\:\:\: \\ $$$$\mathrm{Determine}\:\mathrm{when}\:\mathrm{equality}\:\mathrm{holds}.\:\:\: \\ $$ Answered by mindispower last updated on 27/Jun/21 $$\left(\mathrm{2}{a}+{b}\right)\left({a}+{b}\right)+\left({b}+\mathrm{2}{c}\right)\left({b}+{c}\right) \\…
Question Number 13498 by sin (x) last updated on 20/May/17 Commented by ajfour last updated on 20/May/17 $${if}\:\bigtriangleup{ABC}\:{is}\:{equilateral}, \\ $$$${Area}\left(\bigtriangleup{ABC}\right)=\mathrm{12}×{Area}\left(\bigtriangleup{AED}\right). \\ $$ Answered by mrW1…
Question Number 13491 by 433 last updated on 20/May/17 $${Test} \\ $$$$\mathrm{1}.\:{Solve}\:{equation} \\ $$$$\left({k}^{\mathrm{2}} −\mathrm{1}\right){x}^{\mathrm{2}} +\left({k}−\mathrm{1}\right){x}+\left({k}+\mathrm{1}\right)=\mathrm{0}\:\:{k}\in\mathbb{R} \\ $$$$\left(\mathrm{30}\right) \\ $$$$\mathrm{2}.\:{Prove} \\ $$$$\frac{{sin}\left({x}\right)}{\mathrm{1}+{cos}\left({x}\right)}=\frac{\mathrm{1}−{cos}\left({x}\right)}{{sin}\left({x}\right)} \\ $$$$\left(\mathrm{35}\right) \\…
Question Number 144563 by mathdanisur last updated on 26/Jun/21 $$\boldsymbol{{log}}_{\mathrm{4}} \left(\boldsymbol{{x}}\right)\:=\:\boldsymbol{{log}}\left(\boldsymbol{{y}}\right)\:=\:\boldsymbol{{log}}_{\mathrm{25}} \left(\boldsymbol{{x}}+\boldsymbol{{y}}\right) \\ $$$${Find}\:\:\frac{{x}}{{y}}\:=\:? \\ $$ Answered by imjagoll last updated on 26/Jun/21 $$\:\mathrm{let}\:\mathrm{log}\:_{\mathrm{4}} \left(\mathrm{x}\right)=\mathrm{log}\:_{\mathrm{10}}…