Question Number 215898 by F_Physics last updated on 21/Jan/25 $${x}^{{n}} +{y}^{{n}} +{z}^{{n}} \:=\:\mathrm{0} \\ $$$${n}\:=\:? \\ $$ Commented by Rasheed.Sindhi last updated on 21/Jan/25 $${For}\:{x}={y}={z}=\mathrm{0},\:{n}\in\mathbb{R}^{+}…
Question Number 215908 by MathematicalUser2357 last updated on 21/Jan/25 $$\mathrm{K}.\mathrm{1} \\ $$$${y}=\frac{\mathrm{3}}{\mathrm{2}}{x}+\mathrm{1} \\ $$$$\begin{array}{|c|c|c|c|c|c|}{{x}}&\hline{{y}}\\{−\mathrm{2}}&\hline{}\\{−\mathrm{1}}&\hline{}\\{\mathrm{0}}&\hline{}\\{\mathrm{1}}&\hline{}\\{\mathrm{2}}&\hline{}\\\hline\end{array}\mathrm{and}\:\mathrm{then}\:\mathrm{plot} \\ $$ Answered by Simurdiera last updated on 21/Jan/25 $$\:\underline{{t}} \underline{\underbrace{…
Question Number 215893 by hardmath last updated on 20/Jan/25 $$\mathrm{Find}: \\ $$$$\left.\mathrm{1}\right)\:\underset{\boldsymbol{\mathrm{x}}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{1}\:−\:\mathrm{cos2x}}{\mathrm{x}^{\mathrm{2}} }\:=\:? \\ $$$$\left.\mathrm{2}\right)\:\underset{\boldsymbol{\mathrm{n}}=\mathrm{1}} {\overset{\boldsymbol{\mathrm{n}}=\infty} {\sum}}\:\frac{\mathrm{n}^{\mathrm{2}} }{\mathrm{3}^{\boldsymbol{\mathrm{n}}} }\:=\:? \\ $$ Answered by mathmax…
Question Number 215889 by hardmath last updated on 20/Jan/25 $$\mathrm{Find}:\:\:\:\underset{\boldsymbol{\mathrm{h}}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\left(\mathrm{x}\:+\:\mathrm{h}\right)^{\mathrm{3}} \:+\:\mathrm{x}^{\mathrm{3}} }{\mathrm{h}}\:=\:? \\ $$ Commented by mr W last updated on 20/Jan/25 $$\rightarrow\frac{\mathrm{2}{x}^{\mathrm{3}} }{\mathrm{0}}\rightarrow\infty…
Question Number 215883 by CrispyXYZ last updated on 20/Jan/25 $$\mathrm{Given}\:{m}>\mathrm{0},\:{n}>\mathrm{0},\:{m}+{n}=\sqrt{{a}}. \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{range}\:\mathrm{of}\:{a}\:\mathrm{such}\:\mathrm{that} \\ $$$$“\left({m}+\frac{\mathrm{1}}{{m}}\right)\left({n}+\frac{\mathrm{1}}{{n}}\right)\:\mathrm{gets}\:\mathrm{its}\:\mathrm{minimum}\:\mathrm{iff}\:{m}={n}''. \\ $$ Commented by MathematicalUser2357 last updated on 21/Jan/25 $$“{iff}''?\:{does}\:{it}\:{mean}\:{ifh}.{cc}? \\…
Question Number 215874 by MathematicalUser2357 last updated on 20/Jan/25 $$\mathrm{If}\:{x}^{\mathrm{2}} +\mathrm{3}{x}+\mathrm{2}={y}^{\mathrm{2}} +\mathrm{5}{y}+\mathrm{8}, \\ $$$$\mathrm{Prove}\:\mathrm{that}\:{x}=\frac{−\mathrm{3}\pm\sqrt{\mathrm{4}{y}^{\mathrm{2}} +\mathrm{20}{y}+\mathrm{33}}}{\mathrm{2}}. \\ $$ Answered by MrGaster last updated on 20/Jan/25 $${x}^{\mathrm{2}}…
Question Number 215837 by MATHEMATICSAM last updated on 19/Jan/25 $$\mathrm{If}\:{b}^{\mathrm{3}} \:+\:{a}^{\mathrm{2}} {c}\:+\:{ac}^{\mathrm{2}} \:=\:\mathrm{3}{abc}\:\mathrm{then}\:\mathrm{prove}\:\mathrm{that} \\ $$$$\mathrm{one}\:\mathrm{root}\:\mathrm{of}\:{ax}^{\mathrm{2}} \:+\:{bx}\:+\:{c}\:=\:\mathrm{0}\:\mathrm{is}\:\mathrm{the}\:\mathrm{square}\: \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{other}\:\mathrm{one}. \\ $$ Answered by MrGaster last updated…
Question Number 215820 by Jubr last updated on 18/Jan/25 Commented by A5T last updated on 18/Jan/25 $$\mathrm{This}\:\mathrm{is}\:\mathrm{not}\:\mathrm{generally}\:\mathrm{true},\:\mathrm{it}\:\mathrm{fails}\:\mathrm{when}\:\mathrm{c}=\mathrm{0}. \\ $$ Answered by Rasheed.Sindhi last updated on…
Question Number 215811 by MATHEMATICSAM last updated on 18/Jan/25 $$\mathrm{If}\:\alpha,\:\beta,\:\gamma,\:\delta\:\mathrm{are}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\: \\ $$$${x}^{\mathrm{4}} \:+\:{x}^{\mathrm{3}} \:+\:{x}^{\mathrm{2}} \:+\:{x}\:+\:\mathrm{1}\:=\:\mathrm{0}\:\mathrm{then}\:\mathrm{find} \\ $$$$\alpha^{\mathrm{2021}} \:+\:\beta^{\mathrm{2021}} \:+\:\gamma^{\mathrm{2021}} \:+\:\delta^{\mathrm{2021}} \:. \\ $$ Answered by…
Question Number 215730 by MathematicalUser2357 last updated on 16/Jan/25 $$\boldsymbol{\mathrm{Determine}}\:\boldsymbol{{a}},\:\boldsymbol{{b}},\:\boldsymbol{{c}}\:\left[\mathrm{Lazy}\:\mathrm{problem}\right] \\ $$$$\mathrm{J181}-\mathrm{2}.\:{x}^{\mathrm{3}} −\mathrm{6}{x}^{\mathrm{2}} +\mathrm{15}{x}−\mathrm{7}=\left({x}+{a}\right)^{\mathrm{3}} +{bx}+{c} \\ $$$$\mathrm{J182}-\left(\mathrm{1}\right)\:{x}^{\mathrm{3}} +{ax}+\mathrm{2}=\left({x}+\mathrm{1}\right)\left({x}^{\mathrm{2}} +{bx}+{c}\right) \\ $$ Answered by A5T last…