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man111
 Post subject: polynomial
PostPosted: Sat, 25 Sep 2010 05:54:52 UTC 
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Find $\ value of k for which all the roots of the equation\\ $\ x^4+4x^3-8x^2+k=0 $\ are real.
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outermeasure
 Post subject: Re: polynomial
PostPosted: Sat, 25 Sep 2010 06:03:15 UTC 
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man111 wrote:
Find value of k for which all the roots of the equation
x^4+4x^3-8x^2+k=0
are real.


Hint: where are the local minima of f(x)=x^4+4x^3-8x^2+k?

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\begin{aligned} Spin(1)&=O(1)=\mathbb{Z}/2&\quad&\text{and}\\ Spin(2)&=U(1)=SO(2)&&\text{are obvious}\\ Spin(3)&=Sp(1)=SU(2)&&\text{by }q\mapsto(\mathop{\mathrm{Im}}\mathbb{H}\ni p\mapsto qp\bar{q})\\ Spin(4)&=Sp(1)\times Sp(1)&&\text{by }(q_1,q_2)\mapsto(\mathbb{H}\ni p\mapsto q_1p\bar{q_2})\\ Spin(5)&=Sp(2)&&\text{by }\mathbb{HP}^1\cong S^4_{round}\hookrightarrow\mathbb{R}^5\\ Spin(6)&=SU(4)&&\text{by the irrep }\Lambda_+\mathbb{C}^4 \end{aligned}
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man111
 Post subject:
PostPosted: Sat, 25 Sep 2010 06:47:00 UTC 
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Local $\ Min is at $\ x = -4,$\ x = 1.\\ local $\ Max$\ is at $\ x=0.
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outermeasure
 Post subject:
PostPosted: Sat, 25 Sep 2010 12:54:57 UTC 
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man111 wrote:
Local Min is at $x = -4[/tex], x = 1.
local Max is at x=0.


And what are the values of f in that case? Can you see why this answers the question?

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\begin{aligned} Spin(1)&=O(1)=\mathbb{Z}/2&\quad&\text{and}\\ Spin(2)&=U(1)=SO(2)&&\text{are obvious}\\ Spin(3)&=Sp(1)=SU(2)&&\text{by }q\mapsto(\mathop{\mathrm{Im}}\mathbb{H}\ni p\mapsto qp\bar{q})\\ Spin(4)&=Sp(1)\times Sp(1)&&\text{by }(q_1,q_2)\mapsto(\mathbb{H}\ni p\mapsto q_1p\bar{q_2})\\ Spin(5)&=Sp(2)&&\text{by }\mathbb{HP}^1\cong S^4_{round}\hookrightarrow\mathbb{R}^5\\ Spin(6)&=SU(4)&&\text{by the irrep }\Lambda_+\mathbb{C}^4 \end{aligned}
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man111
 Post subject:
PostPosted: Sat, 25 Sep 2010 17:54:18 UTC 
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$\ yes i have found the solution....\\ f(0) = k \\ f(1) = k-3\\ $ \ and$\ by using drawing The rough graph of f(x).we get\\ k\epsilon\ [0,3].
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