Let $ f $ be a continuous function in $ [a, b] $ show that.

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Let $ f $ be a continuous function in $ [a, b] $ and $ displaystyle int_a ^ b f (t) dt neq0 $. Prove that at any point $ k in (0,1) $ there exists a number $ c in (a, b) $ such that $ displaystyle int_a ^ c f (t) dt = k int_a^b f(t) dt $.

I had the intention of solving it by means of the mean value theorem, for example considering that,
$$F(x)=int_a^xf(t)dtint_a^bf(t)dt$$
but I have a problem, because I get that,
$$int_a^cf(t)dt=frac1(b-a)f(c)left[int_a^bf(t)dtright]^2$$

calculus real-analysis integration

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edited Jul 29 at 17:53

José Carlos Santos

112k1696173

asked Jul 29 at 17:47

Santiago Seeker

577

add a comment | 

up vote
1
down vote

favorite

Let $ f $ be a continuous function in $ [a, b] $ and $ displaystyle int_a ^ b f (t) dt neq0 $. Prove that at any point $ k in (0,1) $ there exists a number $ c in (a, b) $ such that $ displaystyle int_a ^ c f (t) dt = k int_a^b f(t) dt $.

I had the intention of solving it by means of the mean value theorem, for example considering that,
$$F(x)=int_a^xf(t)dtint_a^bf(t)dt$$
but I have a problem, because I get that,
$$int_a^cf(t)dt=frac1(b-a)f(c)left[int_a^bf(t)dtright]^2$$

calculus real-analysis integration

share|cite|improve this question

edited Jul 29 at 17:53

José Carlos Santos

112k1696173

asked Jul 29 at 17:47

Santiago Seeker

577

add a comment | 

up vote
1
down vote

favorite

up vote
1
down vote

favorite

Let $ f $ be a continuous function in $ [a, b] $ and $ displaystyle int_a ^ b f (t) dt neq0 $. Prove that at any point $ k in (0,1) $ there exists a number $ c in (a, b) $ such that $ displaystyle int_a ^ c f (t) dt = k int_a^b f(t) dt $.

I had the intention of solving it by means of the mean value theorem, for example considering that,
$$F(x)=int_a^xf(t)dtint_a^bf(t)dt$$
but I have a problem, because I get that,
$$int_a^cf(t)dt=frac1(b-a)f(c)left[int_a^bf(t)dtright]^2$$

calculus real-analysis integration

share|cite|improve this question

edited Jul 29 at 17:53

José Carlos Santos

112k1696173

asked Jul 29 at 17:47

Santiago Seeker

577

Let $ f $ be a continuous function in $ [a, b] $ and $ displaystyle int_a ^ b f (t) dt neq0 $. Prove that at any point $ k in (0,1) $ there exists a number $ c in (a, b) $ such that $ displaystyle int_a ^ c f (t) dt = k int_a^b f(t) dt $.

I had the intention of solving it by means of the mean value theorem, for example considering that,
$$F(x)=int_a^xf(t)dtint_a^bf(t)dt$$
but I have a problem, because I get that,
$$int_a^cf(t)dt=frac1(b-a)f(c)left[int_a^bf(t)dtright]^2$$

calculus real-analysis integration

share|cite|improve this question

edited Jul 29 at 17:53

José Carlos Santos

112k1696173

asked Jul 29 at 17:47

Santiago Seeker

577

share|cite|improve this question

share|cite|improve this question

edited Jul 29 at 17:53

José Carlos Santos

112k1696173

edited Jul 29 at 17:53

José Carlos Santos

112k1696173

edited Jul 29 at 17:53

José Carlos Santos

112k1696173

112k1696173

asked Jul 29 at 17:47

Santiago Seeker

577

asked Jul 29 at 17:47

Santiago Seeker

577

asked Jul 29 at 17:47

Santiago Seeker

577

577

add a comment | 

add a comment | 

1 Answer
1

active

oldest

votes

up vote
8
down vote



accepted

Let $F(x)=int_a^xf(t),mathrm dt$. Then $F(a)=0$ and $F(b)=int_a^bf(t),mathrm dt$. Since $kint_a^bf(t),mathrm dt$ lies between $F(a)$ and $F(b)$, there must be a $cin(a,b)$ such that$$F(c)left(=int_a^cf(t),mathrm dtright)=kint_a^bf(t),mathrm dt,$$by the Intermediate Value Theorem.

share|cite|improve this answer

edited Jul 29 at 18:24

Alex Provost

15k22147

answered Jul 29 at 17:51

José Carlos Santos

112k1696173

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Excuse a question, how can I know that $ k in (0,1) $ because, it's supposed to have an average interpretation,
  – Santiago Seeker
  Jul 29 at 18:05
  
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  I don't understand this comment. In your question, it was assumed that $kin(0,1)$ and I used that assumption.
  – José Carlos Santos
  Jul 29 at 18:07
  

  
  
  
  

add a comment | 

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1 Answer
1

active

oldest

votes

1 Answer
1

active

oldest

votes

active

oldest

votes

active

oldest

votes

up vote
8
down vote



accepted

Let $F(x)=int_a^xf(t),mathrm dt$. Then $F(a)=0$ and $F(b)=int_a^bf(t),mathrm dt$. Since $kint_a^bf(t),mathrm dt$ lies between $F(a)$ and $F(b)$, there must be a $cin(a,b)$ such that$$F(c)left(=int_a^cf(t),mathrm dtright)=kint_a^bf(t),mathrm dt,$$by the Intermediate Value Theorem.

share|cite|improve this answer

edited Jul 29 at 18:24

Alex Provost

15k22147

answered Jul 29 at 17:51

José Carlos Santos

112k1696173

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Excuse a question, how can I know that $ k in (0,1) $ because, it's supposed to have an average interpretation,
  – Santiago Seeker
  Jul 29 at 18:05
  
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  I don't understand this comment. In your question, it was assumed that $kin(0,1)$ and I used that assumption.
  – José Carlos Santos
  Jul 29 at 18:07
  

  
  
  
  

add a comment | 

up vote
8
down vote



accepted

Let $F(x)=int_a^xf(t),mathrm dt$. Then $F(a)=0$ and $F(b)=int_a^bf(t),mathrm dt$. Since $kint_a^bf(t),mathrm dt$ lies between $F(a)$ and $F(b)$, there must be a $cin(a,b)$ such that$$F(c)left(=int_a^cf(t),mathrm dtright)=kint_a^bf(t),mathrm dt,$$by the Intermediate Value Theorem.

share|cite|improve this answer

edited Jul 29 at 18:24

Alex Provost

15k22147

answered Jul 29 at 17:51

José Carlos Santos

112k1696173

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Excuse a question, how can I know that $ k in (0,1) $ because, it's supposed to have an average interpretation,
  – Santiago Seeker
  Jul 29 at 18:05
  
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  I don't understand this comment. In your question, it was assumed that $kin(0,1)$ and I used that assumption.
  – José Carlos Santos
  Jul 29 at 18:07
  

  
  
  
  

add a comment | 

up vote
8
down vote



accepted

up vote
8
down vote



accepted

Let $F(x)=int_a^xf(t),mathrm dt$. Then $F(a)=0$ and $F(b)=int_a^bf(t),mathrm dt$. Since $kint_a^bf(t),mathrm dt$ lies between $F(a)$ and $F(b)$, there must be a $cin(a,b)$ such that$$F(c)left(=int_a^cf(t),mathrm dtright)=kint_a^bf(t),mathrm dt,$$by the Intermediate Value Theorem.

share|cite|improve this answer

edited Jul 29 at 18:24

Alex Provost

15k22147

answered Jul 29 at 17:51

José Carlos Santos

112k1696173

Let $F(x)=int_a^xf(t),mathrm dt$. Then $F(a)=0$ and $F(b)=int_a^bf(t),mathrm dt$. Since $kint_a^bf(t),mathrm dt$ lies between $F(a)$ and $F(b)$, there must be a $cin(a,b)$ such that$$F(c)left(=int_a^cf(t),mathrm dtright)=kint_a^bf(t),mathrm dt,$$by the Intermediate Value Theorem.

share|cite|improve this answer

edited Jul 29 at 18:24

Alex Provost

15k22147

answered Jul 29 at 17:51

José Carlos Santos

112k1696173

share|cite|improve this answer

share|cite|improve this answer

edited Jul 29 at 18:24

Alex Provost

15k22147

edited Jul 29 at 18:24

Alex Provost

15k22147

edited Jul 29 at 18:24

Alex Provost

15k22147

15k22147

answered Jul 29 at 17:51

José Carlos Santos

112k1696173

answered Jul 29 at 17:51

José Carlos Santos

112k1696173

answered Jul 29 at 17:51

José Carlos Santos

112k1696173

112k1696173

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Excuse a question, how can I know that $ k in (0,1) $ because, it's supposed to have an average interpretation,
  – Santiago Seeker
  Jul 29 at 18:05
  
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  I don't understand this comment. In your question, it was assumed that $kin(0,1)$ and I used that assumption.
  – José Carlos Santos
  Jul 29 at 18:07
  

  
  
  
  

add a comment | 

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Excuse a question, how can I know that $ k in (0,1) $ because, it's supposed to have an average interpretation,
  – Santiago Seeker
  Jul 29 at 18:05
  
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  I don't understand this comment. In your question, it was assumed that $kin(0,1)$ and I used that assumption.
  – José Carlos Santos
  Jul 29 at 18:07
  

  
  
  
  

Excuse a question, how can I know that $ k in (0,1) $ because, it's supposed to have an average interpretation,
– Santiago Seeker
Jul 29 at 18:05

Excuse a question, how can I know that $ k in (0,1) $ because, it's supposed to have an average interpretation,
– Santiago Seeker
Jul 29 at 18:05

2

2

I don't understand this comment. In your question, it was assumed that $kin(0,1)$ and I used that assumption.
– José Carlos Santos
Jul 29 at 18:07

I don't understand this comment. In your question, it was assumed that $kin(0,1)$ and I used that assumption.
– José Carlos Santos
Jul 29 at 18:07

add a comment | 

 

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