Menu Close

prove-that-any-real-root-of-the-equation-x-6n-4x-2n-4-n-N-0-verify-gt-2-1-2n-




Question Number 150883 by mathdanisur last updated on 16/Aug/21
prove that any real root š›‚ of the  equation:  x^(6n)  = 4x^(2n)  + 4 ; nāˆˆN-{0}  verify:  āˆ£š›‚āˆ£ > (2)^(1/(2n))
$$\mathrm{prove}\:\mathrm{that}\:\mathrm{any}\:\mathrm{real}\:\mathrm{root}\:\boldsymbol{\alpha}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{equation}:\:\:\mathrm{x}^{\mathrm{6}\boldsymbol{\mathrm{n}}} \:=\:\mathrm{4x}^{\mathrm{2}\boldsymbol{\mathrm{n}}} \:+\:\mathrm{4}\:;\:\mathrm{n}\in\mathbb{N}-\left\{\mathrm{0}\right\} \\ $$$$\mathrm{verify}:\:\:\mid\boldsymbol{\alpha}\mid\:>\:\sqrt[{\mathrm{2}\boldsymbol{\mathrm{n}}}]{\mathrm{2}} \\ $$
Answered by mindispower last updated on 16/Aug/21
X^(6n) āˆ’4X^(4n) āˆ’4=0  X^(2n) =y>0  ā‡’y^3 āˆ’4y^2 āˆ’4=0  f(y)=y^3 āˆ’4y^2 āˆ’4  fā€²(y)=y(3y^2 āˆ’8),fā€²(y)>0,y>2(āˆš(2/3))=Ī²  f(0)=āˆ’4,f(Ī²)<f(0),f(x)<0,āˆ€xāˆˆ[0,Ī²[  f(2)=āˆ’12  ā‡’f(y)=0,y>2  ā‡’āˆ£yāˆ£=y>2  āˆ£X^(2n) āˆ£>2ā‡’āˆ£Xāˆ£>(2)^(1/(2n))
$${X}^{\mathrm{6}{n}} āˆ’\mathrm{4}{X}^{\mathrm{4}{n}} āˆ’\mathrm{4}=\mathrm{0} \\ $$$${X}^{\mathrm{2}{n}} ={y}>\mathrm{0} \\ $$$$\Rightarrow{y}^{\mathrm{3}} āˆ’\mathrm{4}{y}^{\mathrm{2}} āˆ’\mathrm{4}=\mathrm{0} \\ $$$${f}\left({y}\right)={y}^{\mathrm{3}} āˆ’\mathrm{4}{y}^{\mathrm{2}} āˆ’\mathrm{4} \\ $$$${f}'\left({y}\right)={y}\left(\mathrm{3}{y}^{\mathrm{2}} āˆ’\mathrm{8}\right),{f}'\left({y}\right)>\mathrm{0},{y}>\mathrm{2}\sqrt{\frac{\mathrm{2}}{\mathrm{3}}}=\beta \\ $$$${f}\left(\mathrm{0}\right)=āˆ’\mathrm{4},{f}\left(\beta\right)<{f}\left(\mathrm{0}\right),{f}\left({x}\right)<\mathrm{0},\forall{x}\in\left[\mathrm{0},\beta\left[\right.\right. \\ $$$${f}\left(\mathrm{2}\right)=āˆ’\mathrm{12} \\ $$$$\Rightarrow{f}\left({y}\right)=\mathrm{0},{y}>\mathrm{2} \\ $$$$\Rightarrow\mid{y}\mid={y}>\mathrm{2} \\ $$$$\mid{X}^{\mathrm{2}{n}} \mid>\mathrm{2}\Rightarrow\mid{X}\mid>\sqrt[{\mathrm{2}{n}}]{\mathrm{2}} \\ $$$$ \\ $$$$ \\ $$$$ \\ $$
Commented by mathdanisur last updated on 16/Aug/21
Thank you Ser, but  4x^(4n)   no 4x^(2n)   ā‡’ y^3  - 4y^2 .? or 4y
$$\mathrm{Thank}\:\mathrm{you}\:\mathrm{Ser},\:\mathrm{but}\:\:\mathrm{4x}^{\mathrm{4}\boldsymbol{\mathrm{n}}} \:\:\mathrm{no}\:\mathrm{4x}^{\mathrm{2}\boldsymbol{\mathrm{n}}} \\ $$$$\Rightarrow\:\mathrm{y}^{\mathrm{3}} \:-\:\mathrm{4y}^{\mathrm{2}} .?\:\mathrm{or}\:\mathrm{4y} \\ $$

Leave a Reply

Your email address will not be published. Required fields are marked *