Question Number 203379 by York12 last updated on 18/Jan/24 $$\mathrm{If}\:{a},{b},{c}\:\in\mathbb{R}^{+} \:\mathrm{with}\:{a}+{b}+{c}=\mathrm{3}\:\mathrm{prove}\:\mathrm{that} \\ $$$$\frac{\mathrm{1}}{{a}^{\mathrm{6}} +{b}^{\mathrm{6}} +\mathrm{3}{c}^{\mathrm{3}} +\mathrm{4}}+\frac{\mathrm{1}}{{b}^{\mathrm{6}} +{c}^{\mathrm{6}} +\mathrm{3}{a}^{\mathrm{3}} +\mathrm{4}}+\frac{\mathrm{1}}{{c}^{\mathrm{6}} +{a}^{\mathrm{6}} +\mathrm{3}{b}^{\mathrm{3}} +\mathrm{4}}\leqslant\frac{\mathrm{3}}{\mathrm{3}+\mathrm{2}\left(\sqrt{{ab}}+\sqrt{{bc}}+\sqrt{{ac}}\right)} \\ $$ Terms…
Question Number 203424 by Calculusboy last updated on 18/Jan/24 Answered by Rasheed.Sindhi last updated on 19/Jan/24 $${x}+{y}=\mathrm{2}^{{x}−{y}} \:…\left({i}\right) \\ $$$$\left({x}+{y}\right)^{{x}−{y}} =\mathrm{2}…\left({ii}\right) \\ $$$$\left({ii}\right)/\left({i}\right):\:\left({x}+{y}\right)^{{x}−{y}−\mathrm{1}} =\mathrm{2}^{\mathrm{1}−{x}+{y}} \\…
Question Number 203357 by Calculusboy last updated on 17/Jan/24 $$\boldsymbol{{For}}\:\boldsymbol{{the}}\:\boldsymbol{{series}}\:\mathrm{5}−\frac{\mathrm{5}}{\mathrm{2}}+\frac{\mathrm{5}}{\mathrm{4}}−\frac{\mathrm{5}}{\mathrm{8}}+\centerdot\centerdot\centerdot+\frac{\left(−\mathrm{1}\right)^{\boldsymbol{{n}}−\mathrm{1}} \mathrm{5}}{\mathrm{2}^{\boldsymbol{{n}}−\mathrm{1}} } \\ $$$$\boldsymbol{{find}}\:\boldsymbol{{an}}\:\boldsymbol{{expression}}\:\boldsymbol{{for}}\:\boldsymbol{{the}}\:\boldsymbol{{sum}}\:\boldsymbol{{of}}\:\boldsymbol{{the}}\:\boldsymbol{{first}} \\ $$$$\boldsymbol{{n}}\:\boldsymbol{{terms}}.\:\boldsymbol{{Also}}\:\boldsymbol{{if}}\:\boldsymbol{{the}}\:\boldsymbol{{series}}\:\boldsymbol{{converges}}, \\ $$$$\boldsymbol{{find}}\:\boldsymbol{{the}}\:\boldsymbol{{sum}}\:\boldsymbol{{to}}\:\infty. \\ $$$$ \\ $$$$ \\ $$ Answered…
Question Number 203352 by Mingma last updated on 17/Jan/24 Answered by witcher3 last updated on 17/Jan/24 $$\mathrm{sin}\left(\mathrm{2x}\right)=\mathrm{0}\Rightarrow\mathrm{x}=\frac{\mathrm{k}\pi}{\mathrm{2}},\mathrm{k}\in\mathbb{Z} \\ $$$$\mathrm{x}^{\mathrm{2}} −\mathrm{5x}+\mathrm{4}=\mathrm{x}^{\mathrm{2}} −\left(\mathrm{1}+\mathrm{4}\right)\mathrm{x}+\left(\mathrm{1}.\mathrm{4}\right)=\mathrm{0}\Rightarrow\mathrm{x}\in\left\{\mathrm{1},\mathrm{4}\:\right\} \\ $$$$\mathrm{X}=\mathbb{R}−\left(\left\{\mathrm{1},\mathrm{4}\right\}\cup\left\{\frac{\mathrm{k}\pi}{\mathrm{2}},\mathrm{k}\in\mathbb{Z}\right\}\right) \\ $$…
Question Number 203353 by Mingma last updated on 17/Jan/24 Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 203351 by Mingma last updated on 17/Jan/24 Answered by witcher3 last updated on 17/Jan/24 $$\mathrm{x}\in\mathrm{E}\cap\mathrm{F}\Leftrightarrow\left(\mathrm{x}\in\mathrm{E}\:\&\mathrm{x}\in\mathrm{F}\right) \\ $$$$\Leftrightarrow\left(\mathrm{x}\in\mathbb{N}\:\&\mathrm{1}\leqslant\mathrm{x}\leqslant\mathrm{15\&}\frac{\mathrm{x}+\mathrm{1}}{\mathrm{2}}\in\mathbb{Z}\right) \\ $$$$\mathrm{x}\in\mathbb{N}\Rightarrow\mathrm{x}\geqslant\mathrm{0}\:\Rightarrow\frac{\mathrm{x}+\mathrm{1}}{\mathrm{2}}\in\mathbb{Z}\Rightarrow\frac{\mathrm{x}+\mathrm{1}}{\mathrm{2}}\geqslant\mathrm{1} \\ $$$$\Leftrightarrow\left\{\mathrm{x}\in\mathbb{N};\mathrm{1}\leqslant\mathrm{x}\leqslant\mathrm{15},\:\mathrm{x}+\mathrm{1}\mid\mathrm{2}\right\},\mathrm{x}=\left\{\mathrm{1},\mathrm{3},\mathrm{5},\mathrm{7},\mathrm{9},\mathrm{11},\mathrm{13},\mathrm{15}\right\} \\ $$…
Question Number 203367 by mr W last updated on 17/Jan/24 $${if}\:\boldsymbol{{a}}+\boldsymbol{{b}}=\mathrm{198},\:{what}\:{is}\:{the}\:{largest} \\ $$$$\boldsymbol{{integer}}\:{root}\:{which}\:{the}\:{equation} \\ $$$$\boldsymbol{{x}}^{\mathrm{2}} +\boldsymbol{{ax}}+\boldsymbol{{b}}=\mathrm{0}\:{may}\:{have}? \\ $$ Answered by ajfour last updated on 18/Jan/24…
Question Number 203329 by Calculusboy last updated on 16/Jan/24 $$\boldsymbol{{find}}\:\boldsymbol{{the}}\:\boldsymbol{{ranges}}\:\boldsymbol{{of}}\:\boldsymbol{{value}}\:\boldsymbol{{of}}\:\boldsymbol{{x}}\:\boldsymbol{{for}}\:\boldsymbol{{which}}\: \\ $$$$\boldsymbol{{series}}\:\boldsymbol{{convergent}}\:\boldsymbol{{or}}\:\boldsymbol{{divergent}} \\ $$$$\underset{\boldsymbol{{n}}=\mathrm{1}} {\overset{\infty} {\boldsymbol{\sum}}}\frac{\left(\boldsymbol{{n}}+\mathrm{1}\right)}{\boldsymbol{{n}}^{\mathrm{3}} }\boldsymbol{{x}}^{\boldsymbol{{n}}} \\ $$ Answered by Mathspace last updated on…
Question Number 203309 by hardmath last updated on 15/Jan/24 $$\mathrm{If}\:\:\:\mathrm{z}_{\mathrm{1}} =\:\mathrm{3}\:+\:\mathrm{3}\sqrt{\mathrm{3}}\:\boldsymbol{\mathrm{i}}\:\:\:\mathrm{and}\:\:\:\mathrm{z}_{\mathrm{2}} =\:-\mathrm{1}\:−\:\sqrt{\mathrm{3}}\:\boldsymbol{\mathrm{i}} \\ $$$$\mathrm{Find}:\:\:\:\frac{\mathrm{z}_{\mathrm{1}} ^{\:\mathrm{3}} }{\mathrm{z}_{\mathrm{2}} ^{\:\mathrm{6}} }\:=\:? \\ $$ Answered by Rasheed.Sindhi last updated…
Question Number 203305 by Ari last updated on 15/Jan/24 Answered by mr W last updated on 15/Jan/24 $${y}={mx}+{c} \\ $$$$\Rightarrow{x}=\frac{{y}}{{m}}−\frac{{c}}{{m}}\:\Rightarrow{f}^{−\mathrm{1}} \left({x}\right)=\frac{{x}}{{m}}−\frac{{c}}{{m}} \\ $$$${f}^{−\mathrm{1}} \left({x}\right)={f}\left({x}\right) \\…