[問題] 分析─請問怎麼取Sup?

[A-gine]

函數序列：

____x_____
f_n(x) = 1+nx^2 ( x除以1+nx^2 ) |x|≦ 1 , x 屬於 |R


請問 Sup | f_n(x) | ＝ ？ ( | f_n(x) |←絕對值 f_n(x) )
|x|≦1


麻煩大大們了>""<

謝謝大家~

--
※ Origin: 楓橋驛站<bbs.cs.nthu.edu.tw>
◆ From: iamagine @218.172.75.143

[天才軒]

Post by A-gine
函數序列：
____x_____
f_n(x) = 1+nx^2 ( x除以1+nx^2 ) |x|≦ 1 , x 屬於 |R
請問 Sup | f_n(x) | ＝ ？ ( | f_n(x) |←絕對值 f_n(x) )
|x|≦1
麻煩大大們了>""<
謝謝大家~

It's easy to see that sup |fn(x)| = sup fn(x)
|x|≦1 [0,1]

For x in (0,1), d/dx fn(x) = (1+nx^2 - 2nx^2)/(1+nx^2)^2

= (1-nx^2)/(1+nx^2)^2

If fn has a local maximum at x, 1=nx^2 => fn(x) = 1/(2√n)

fn(0) = 0 < fn(x) and fn(1) = 1/(1+n) ≦ 1/(2√n) = fn(x);

hence, sup |fn(x)| = 1/(2√n).

--

弄清π是無理數這件事可能是根本沒有實際用處的

但是如果我們能弄清楚那麼肯定就不能容忍不去設法把它弄清楚

 ──E.C.Titchmarsh
--
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 http://cd.twbbs.org◥  ＊ Author: Fresh ★ From: 220.138.61.235   ◤

[A-gine]

Post by A-gine
函數序列：
____x_____
f_n(x) = 1+nx^2 ( x除以1+nx^2 ) |x|≦ 1 , x 屬於 |R
請問 Sup | f_n(x) | ＝ ？ ( | f_n(x) |←絕對值 f_n(x) )
|x|≦1
麻煩大大們了>""<
謝謝大家~

It's easy to see that sup |fn(x)| = sup fn(x)
 |x|≦1 [0,1]

^^^^^^^^^^^^^^^^^^^^^^^^^^
這個看不太出來，麻煩大大提示>""<

For x in (0,1), d/dx fn(x) = (1+nx^2 - 2nx^2)/(1+nx^2)^2
= (1-nx^2)/(1+nx^2)^2
If fn has a local maximum at x, 1=nx^2 => fn(x) = 1/(2√n)
fn(0) = 0 < fn(x) and fn(1) = 1/(1+n) ≦ 1/(2√n) = fn(x);
hence, sup |fn(x)| = 1/(2√n).

--
※ Origin: 楓橋驛站<bbs.cs.nthu.edu.tw>
◆ From: iamagine @218.172.78.136

[天才軒]

Post by A-gine

It's easy to see that sup |fn(x)| = sup fn(x)
 |x|≦1 [0,1]

^^^^^^^^^^^^^^^^^^^^^^^^^^
這個看不太出來，麻煩大大提示>""<

For x in (0,1), d/dx fn(x) = (1+nx^2 - 2nx^2)/(1+nx^2)^2
= (1-nx^2)/(1+nx^2)^2
If fn has a local maximum at x, 1=nx^2 => fn(x) = 1/(2√n)
fn(0) = 0 < fn(x) and fn(1) = 1/(1+n) ≦ 1/(2√n) = fn(x);
hence, sup |fn(x)| = 1/(2√n).

Since |fn| is contionous and [-1,1] is compact, there exist pn in [-1,1]

such that |fn(pn)| = sup |fn|.

Since |fn| is even function, WLOG we may assume pn≧0;

hence, sup|fn| = sup |fn| = sup fn
[-1,1] [0,1] [0,1]

--

弄清π是無理數這件事可能是根本沒有實際用處的

但是如果我們能弄清楚那麼肯定就不能容忍不去設法把它弄清楚

 ──E.C.Titchmarsh
--
● ˙ ˙ ◢▇◣ ◢▇◣ ▇ ▇ █▇◣ █▇◣ █▇◣ ◢▇◣ █▇▉ ／
˙ ╲ █ █ █ █ █ ▉ █ ▉ █▆ █▆▉ ▉▉▉ ★
◢ ★˙ ◥█◤ ◥█◤ ◥█◤ ██◤ █◥▆ █▆◤ █ ▉ ▉▉▉˙ ◣
 ＣＤＢＢＳ 中正築夢園BBS站 歡迎蒞臨參觀 cd.twbbs.org (140.123.20.230)  
 http://cd.twbbs.org◥  ＊ Author: Fresh ★ From: 220.138.59.177   ◤