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Question Number 150903 Answers: 1 Comments: 0
$${let}\:{x},{y}>\mathrm{0}\:,\:{n}\in\mathbb{N}, \\ $$$${show}\:{that}\:\left({x}+{y}\right)^{{n}} \leqslant\mathrm{2}^{{n}−\mathrm{1}} \left({x}^{{n}} +{y}^{{n}} \right).. \\ $$
Question Number 150895 Answers: 1 Comments: 1
Question Number 150889 Answers: 1 Comments: 0
Question Number 150887 Answers: 1 Comments: 4
Question Number 150886 Answers: 0 Comments: 0
$$\:\varphi\left(\mathrm{n}\right)=\varphi\left(\mathrm{n}+\mathrm{1}\right)=\varphi\left(\mathrm{n}+\mathrm{2}\right)\:\:\mathrm{n}\in\mathrm{N}\:\mathrm{n}_{\mathrm{min}} =? \\ $$$$\varphi\left(\mathrm{n}\right)−\:\mathrm{Euler}\:\mathrm{funcsion} \\ $$$$ \\ $$$$ \\ $$$$ \\ $$
Question Number 150885 Answers: 0 Comments: 0
Question Number 150884 Answers: 1 Comments: 0
Question Number 150883 Answers: 1 Comments: 0
$$\mathrm{prove}\:\mathrm{that}\:\mathrm{any}\:\mathrm{real}\:\mathrm{root}\:\boldsymbol{\alpha}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{equation}:\:\:\mathrm{x}^{\mathrm{6}\boldsymbol{\mathrm{n}}} \:=\:\mathrm{4x}^{\mathrm{2}\boldsymbol{\mathrm{n}}} \:+\:\mathrm{4}\:;\:\mathrm{n}\in\mathbb{N}-\left\{\mathrm{0}\right\} \\ $$$$\mathrm{verify}:\:\:\mid\boldsymbol{\alpha}\mid\:>\:\sqrt[{\mathrm{2}\boldsymbol{\mathrm{n}}}]{\mathrm{2}} \\ $$
Question Number 150881 Answers: 1 Comments: 0
$$\mathrm{y}\:=\:\left(\frac{\mathrm{a}}{\mathrm{b}}\right)^{\boldsymbol{\mathrm{x}}} \centerdot\:\left(\frac{\mathrm{b}}{\mathrm{x}}\right)^{\boldsymbol{\mathrm{a}}} \centerdot\:\left(\frac{\mathrm{x}}{\mathrm{a}}\right)^{\boldsymbol{\mathrm{b}}} \:\Rightarrow\:\mathrm{y}\:^{'} \:=\:? \\ $$
Question Number 150880 Answers: 2 Comments: 0
$$\mathrm{if}\:\:\mathrm{a};\mathrm{b};\mathrm{c}>\mathrm{0}\:\:\mathrm{and}\:\:\mathrm{a}+\mathrm{b}+\mathrm{c}=\mathrm{1} \\ $$$$\mathrm{find}\:\boldsymbol{\mathrm{min}}\left(\frac{\mathrm{1}}{\mathrm{a}}\:+\:\frac{\mathrm{9}}{\mathrm{b}}\:+\:\frac{\mathrm{25}}{\mathrm{c}}\right)\:=\:? \\ $$$$\left.\mathrm{a}\left.\right)\left.\mathrm{7}\left.\mathrm{3}\left.\:\:\:\mathrm{b}\right)\mathrm{75}\:\:\:\mathrm{c}\right)\mathrm{105}\:\:\:\mathrm{d}\right)\mathrm{81}\:\:\:\mathrm{e}\right)\mathrm{83} \\ $$
Question Number 150876 Answers: 0 Comments: 0
Question Number 150865 Answers: 0 Comments: 0
Im{f'(z)} = 6x(2y-1) and f(0) = 3 - 2i, f(1) = 6 - 5i. Find f(2 + i)
Question Number 150864 Answers: 1 Comments: 0
Question Number 150862 Answers: 0 Comments: 0
Question Number 150861 Answers: 1 Comments: 0
$${Find}\:\:{the}\:\:{solution}\:\:{of}\:\:: \\ $$$$\left\{_{{x}^{\mathrm{2}} +\mathrm{3}{xy}+\mathrm{2}{y}^{\mathrm{2}} −\mathrm{4}\:=\:\mathrm{0}} ^{\mathrm{2}{x}^{\mathrm{2}} −\mathrm{2}{xy}−\mathrm{3}{y}^{\mathrm{2}} +\mathrm{7}\:=\:\mathrm{0}} \right. \\ $$$${Please}\:\:{show}\:\:{your}\:\:{working}... \\ $$
Question Number 150859 Answers: 0 Comments: 0
Question Number 150853 Answers: 0 Comments: 2
$$\mathrm{log}_{\mathrm{2021}} \:\sqrt{\mathrm{x}\::\:\sqrt{\mathrm{x}\::\:\sqrt{\mathrm{x}\::..}}}\:=\:\mathrm{674} \\ $$$$\mathrm{find}\:\:\boldsymbol{\mathrm{x}}=? \\ $$
Question Number 150852 Answers: 0 Comments: 0
$$\mathrm{x};\mathrm{y};\mathrm{z}>\mathrm{0}\:\:\mathrm{and}\:\:\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} +\mathrm{z}^{\mathrm{2}} =\mathrm{3}\:\:\mathrm{prove}\:\mathrm{that} \\ $$$$\mathrm{xyz}\:\leqslant\:\left(\frac{\mathrm{x}\:+\:\mathrm{y}\:+\:\mathrm{z}\:-\:\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{2}} \:\leqslant\:\mathrm{1} \\ $$
Question Number 150841 Answers: 2 Comments: 0
$$\underset{\:\mathrm{0}} {\overset{\:\infty} {\int}}\:\frac{\mathrm{ln}\left(\mathrm{1}+\mathrm{a}^{\mathrm{2}} \mathrm{x}^{\mathrm{2}} \right)}{\mathrm{1}+\mathrm{b}^{\mathrm{2}} \mathrm{x}^{\mathrm{2}} }\:\mathrm{dx}\:=\:? \\ $$
Question Number 150840 Answers: 1 Comments: 0
$$\underset{\:\mathrm{0}} {\overset{\:\infty} {\int}}\:\frac{\mathrm{arctan}\left(\mathrm{x}\right)}{\mathrm{x}\left(\mathrm{x}^{\mathrm{2}} +\mathrm{x}+\mathrm{1}\right)}\:\mathrm{dx}\:=\:? \\ $$
Question Number 150839 Answers: 2 Comments: 0
$$\mathrm{e}^{\boldsymbol{\mathrm{x}}} \:+\:\mathrm{y}\:=\:\mathrm{x}^{\mathrm{2}} \mathrm{y}^{\mathrm{2}} \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{expression}\:\mathrm{for}\:\:\frac{\mathrm{dy}}{\mathrm{dx}} \\ $$
Question Number 150838 Answers: 1 Comments: 0
Question Number 150828 Answers: 2 Comments: 0
$$ \\ $$$$\:\int_{\mathrm{0}} ^{\:\infty} \frac{\:{sin}^{\:\mathrm{2}} \left({x}\:\right)}{{x}\sqrt{{x}}}\:{dx}\overset{?} {=}\:\sqrt{\pi} \\ $$
Question Number 150827 Answers: 0 Comments: 0
Question Number 150807 Answers: 5 Comments: 0
$$\mathrm{For}\:\mathrm{matris}\:\mathrm{solution}: \\ $$$$\begin{cases}{\mathrm{2x}\:-\:\mathrm{3y}\:=\:\mathrm{8}}\\{\mathrm{x}\:+\:\mathrm{5y}\:=\:-\:\mathrm{9}}\end{cases} \\ $$
Question Number 150946 Answers: 0 Comments: 0
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