スキップしてメイン コンテンツに移動

Rolle’s theorem

Introduction

This post is written Rolle’s theorem.
The mean-value theorem is proved by Rolle’s theorem.
I will write Mean-value theorem at a later.
I introduce Maximum principle because proving Rolle’s theorem need Maximum principle.

Maximum principle

It is very easy.
f is continuous function on bounded closed interval.\(\implies\)**
f have max value.**

Proof

This proof is difficult.
I write this proof in other posts.
Maximum Principle

Rolle’s theorem

f is continuous function on [a,b] and differentiable function on (a,b).
\[f(a) = f(b) \implies \exists ~~c ~~s.t~~ f'(c) = 0 , a<c<b\]

Proof

  • f(x) is constant function
    \[\forall c \in (a,b) , f'(c) = 0\]
  • else
when \(\exists t ~~s.t~~f(a) < f(t)\), \(\exists c ~~s.t~~ \max f(x) = f(c)\) by Maximum principle
I proof \(f'(c)=0\)
f is differentiable on \(x = c\) and \(f(c) >= f(c+h)\).
Thus
\[f'(c) = \lim_{h \rightarrow +0} \frac{f(c+h) - f(c)}{h} \leq 0\]
\[f'(c) = \lim_{h \rightarrow -0} \frac{f(c+h) - f(c)}{h} \geq 0\]
Therefore \[0 \leq \lim_{h \rightarrow -0} \frac{f(c+h) - f(c)}{h} =f'(c) = \lim_{h \rightarrow +0} \frac{f(c+h) - f(c)}{h} \leq 0\]
\[f'(c)=0\]
when\(\exists t ~~s.t f(a)>f(t)\), proof is same.

Image

enter image description here
when \(f(3)=f(5)\) , function have to turn.
This Turning point is c!!

Conclusion

Rolle’s theorem is used proof of the mean-value theorem.
I write mean-value theorem on other posts.
Mean-Value Theorem

Reference

https://mathtrain.jp/rolle
Labels:

コメント

コメントを投稿

このブログの人気の投稿

離散フーリエ変換

Introduction English ver 今日は離散フーリエ変換について書きます。 現在、シグナル解析についてのpdfを作成中です。このpdfは github で公開中です。 シグナル解析は、courseraのレクチャーで勉強中です。 ここ にリンクを貼っておきます。 pdfは随時更新中です。
2 件のコメント
続きを読む

heap structure

Introduction 日本語 ver Today, I will write about heap structure. The heap structure is one of the data structure. My reason of studying heap structure is that I joined seminar of Ritsumeikan Univ. I used this  slide  in seminar of Ritsumeikan Univ. Overview data structure binary tree heap Implementation heap sort Data structure I will explain about data structure before explaining about heap. Data structure is how to keep data. Data structure is selected on the basis of operation which you want to.  Heap belong to data structure called priority queue. priority queue have purpose which  add data pick up minimum data (and remove)  Binary Tree Let, E and V are sets. The element $e \in E$ is called edge. The element $v \in V$ is called node. g:E->V×V is map to V × V from E. (E,V,g) is called graph. For example, The arrows are edge. The circles are node. This is expressed map. It is possible to go to node 3 from 1....
コメントを投稿
続きを読む

LIMIT and user variable in MysQL

Problem 日本語 ver I want to use this query in PROCEDURE. SET @user_variable = FLOOR(RAND()*10); INSERT INTO  Table_name (columns_name) VALUES (1) FROM Table_name ORDER BY RAND() LIMIT 1 OFFSET @user_variable; This query want to substitute 1 for the @user_variable 'th line of columns_name, but this query does not work. A cause is that LIMIT and OFFSET does not user variable. Solution SET @user_variable = FLOOR(RAND()*10); PREPARE SET_STMT FROM 'INSERT INTO  Table_name (columns_name) VALUES (1) FROM Table_name ORDER BY RAND() LIMIT 1 OFFSET ?;';  EXECUTE SET_STMT USING @user_variable; Conclusion, if you want to use user variable with LIMIT or OFFSET, use PREPARE STATMENT. Reference http://techtipshoge.blogspot.com/2011/10/limit.html
2 件のコメント
続きを読む