Question Number 215991 by F_Physics last updated on 24/Jan/25 $$\underset{{x}=\mathrm{1}} {\overset{{y}} {\sum}}{x}^{{n}} =\mathrm{0} \\ $$$${n}=? \\ $$ Commented by mr W last updated on 25/Jan/25…
Question Number 215973 by MATHEMATICSAM last updated on 23/Jan/25 $$\left({a}\:+\:{b}\right)\:\propto\:{c}\:\mathrm{and}\:\left({b}\:+\:{c}\right)\:\propto\:{a}.\:\mathrm{Prove}\:\mathrm{that} \\ $$$$\left({c}\:+\:{a}\right)\:\propto\:{b}. \\ $$ Answered by Rasheed.Sindhi last updated on 24/Jan/25 $$\left({a}\:+\:{b}\right)\:\propto\:{c}\:\mathrm{and}\:\left({b}\:+\:{c}\right)\:\propto\:{a}.\:\mathrm{Prove}\:\mathrm{that} \\ $$$$\left({c}\:+\:{a}\right)\:\propto\:{b}. \\…
Question Number 215957 by LG last updated on 22/Jan/25 $${b}^{\mathrm{2}} −\mathrm{4}{ac} \\ $$ Commented by MathematicalUser2357 last updated on 23/Jan/25 $${x}=\frac{−{b}\pm\sqrt{\boldsymbol{{b}}^{\mathrm{2}} −\mathrm{4}\boldsymbol{{ac}}}}{\mathrm{2}{a}} \\ $$ Commented…
Question Number 215898 by F_Physics last updated on 21/Jan/25 $${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…