[微分] 求救求救...都是我的問題!??

[victor7935 (victor)]

1. Show that

        x - (x^3)/6 < sinx  for all x > 0

    微1次後

   f(x) = sinx -x + (1/6)x^3

   f'(x)= cosx -1 + (1/2)x^2

   然後.....

2.
   Suppose that f is continuous on [a,b] and f(a)=f(b).

   Show that f has at least one critical point in (a,b).



3.
   (a)

   Give an example of a nonconstant function that takes on both its

   absolute max and absolute min on every interval.

   (b)

    Give an example of a nonconstant function that has an iinfinite

   number of distinct local max and an infinite number of distinct

  local  min.

4.Use

   1 - (1/2)x^2 < cosx < 1- (1/2)x^2 + (1/24)x^4    for all x>0.

   to estimate cos 6度(the x above is in radians.)



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※ 發信站: 批踢踢實業坊(ptt.cc)
◆ From: 140.123.221.147
→ Qmmm:1.x=0為C.P.,f(0)=0為 absolute minmum.
→ Qmmm:f(0) < f(x)  , for all x>0
推 TaiwanFlight:1. let cosx=-1 then f'(x)=(1/2)x^2
→ TaiwanFlight:>0 for all x
→ TaiwanFlight:so f(x)遞增
→ Qmmm:2.少了f在(a,b)可微的條件
→ TaiwanFlight:2. by Thm , true
→ TaiwanFlight:critical point是什麼點?
推 TaiwanFlight:2.f在[a,b]連續且f(x)=f(b)
→ TaiwanFlight:則f至少有一個臨界點在(a,b)
→ TaiwanFlight:f不需要可微
→ TaiwanFlight:                f(a)
→ victor7935:那這樣是要有哪個定義證?!!
→ victor7935:          用
※ 編輯: victor7935      來自: 140.123.221.147      (11/10 00:20)
→ Qmmm: 2. 考慮f(x)=|x| ,f is cont. everywhere
→ Qmmm:f(1)=f(-1) 但f在(-1,1)不可微
※ 編輯: victor7935      來自: 140.123.221.147      (11/10 00:22)
推 TaiwanFlight:2. set f(x)=1 ,x no critical point
→ victor7935:所以不可微那點就是critical point!!
→ victor7935:是降嗎?
→ Qmmm:嗯嗯~~ 不存在點也是臨界點 抱歉忘了^^"
→ victor7935:那就是分2種討論!?  1.可微..by MVT
→ victor7935:因為f(a)=f(b) ..f'(c)=0 ..2不可微.
→ victor7935:之點及是臨界點...這樣寫ok?
推 TaiwanFlight:3.(a)  let f(x)=sin(x)
推 TaiwanFlight:3.(b) let f(x)=sec(x);(-pi/2,pi/2)
推 TaiwanFlight:4. x=0.104719667帶入
→ muxiv       : 因為f(a)=f(b) https://noxiv.com