作者sato186 (银色轰炸机)
看板Math
标题Re: [微分] 勘根定理
时间Mon Mar 7 00:04:29 2011
※ 引述《immortal8305 (immortal)》之铭言：
: ※ [本文转录自 trans_math 看板 #1DSwZ0RD ]
: 作者: immortal8305 (immortal) 看板: trans_math
: 标题: [微分] 勘根定理
: 时间: Sun Mar 6 23:31:09 2011
: Let p(x) be a polynomial of odd degree with real coefficient.Then equation
: p(x)=0 has at least one real root.prove this.
n n-1
Set p(x) ＝ a x + a x + ... + a x + a has odd degree.
n n-1 1 0
WLOG, assume that a ＞ 0. Let
M ＝ max {|a |, |a |, |a |, |a |, 1.}.
n
n n-1 1 0
1
Since a ＞ 0, there exists a positive integer N such that
a ＞ ── .
n
n N
n n-1 n-2
So p(kM)
≧ a (kM) – M(kM) – M(kM) – ... – M(kM) – M ＝
n
n n
n n n k -1 n n n k -1 1 n
a k M – M ────
≧ a k M – M ──── ＞ (a – ──) k
n k-1 k N k N
for
k-1 ≧ N. So
p( (N+1)M ) ＞ 0. Similarly,
p( -(N+1)M ) ＜ 0.
By intermediate value theorem, p(c) ＝ 0 for some c in
( -(N+1)M, (N+1)M ).
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1F：推 immortal8305:谢谢 03/07 12:58