Root of $tan x- x=6$ [closed]

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In my book a problem is given as follows:

Show that the equation $tan x- x=6$ has one and only one root in the interval $(frac-pi2,fracpi2 )$.

This exercise problem is given after discussing rolle's theorem and mean value theorem so I guess it is an application of these theorems.

calculus real-analysis derivatives

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edited Jul 25 at 13:36

Parcly Taxel

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asked Jul 25 at 13:30

jiren

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closed as off-topic by John Ma, Jendrik Stelzner, John B, Shailesh, Carl Mummert Aug 8 at 1:15

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – John Ma, Jendrik Stelzner, John B, Shailesh, Carl Mummert

If this question can be reworded to fit the rules in the help center, please edit the question.

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Yes, you're right about using the mean value theorem! Now suppose there are two such points where $f(x) = tan x - x = 6$. Can you use the MVT to arrive at a contradiction?
  – Cataline
  Jul 25 at 13:35
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Could you please include a more specific source - which book and problem are you looking at?
  – Carl Mummert
  Aug 8 at 1:15
  

  
  
  
  

add a comment | 

up vote
0
down vote

favorite

1

In my book a problem is given as follows:

Show that the equation $tan x- x=6$ has one and only one root in the interval $(frac-pi2,fracpi2 )$.

This exercise problem is given after discussing rolle's theorem and mean value theorem so I guess it is an application of these theorems.

calculus real-analysis derivatives

share|cite|improve this question

edited Jul 25 at 13:36

Parcly Taxel

33.5k136588

asked Jul 25 at 13:30

jiren

284

closed as off-topic by John Ma, Jendrik Stelzner, John B, Shailesh, Carl Mummert Aug 8 at 1:15

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – John Ma, Jendrik Stelzner, John B, Shailesh, Carl Mummert

If this question can be reworded to fit the rules in the help center, please edit the question.

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Yes, you're right about using the mean value theorem! Now suppose there are two such points where $f(x) = tan x - x = 6$. Can you use the MVT to arrive at a contradiction?
  – Cataline
  Jul 25 at 13:35
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Could you please include a more specific source - which book and problem are you looking at?
  – Carl Mummert
  Aug 8 at 1:15
  

  
  
  
  

add a comment | 

up vote
0
down vote

favorite

1

up vote
0
down vote

favorite

1

1

In my book a problem is given as follows:

Show that the equation $tan x- x=6$ has one and only one root in the interval $(frac-pi2,fracpi2 )$.

This exercise problem is given after discussing rolle's theorem and mean value theorem so I guess it is an application of these theorems.

calculus real-analysis derivatives

share|cite|improve this question

edited Jul 25 at 13:36

Parcly Taxel

33.5k136588

asked Jul 25 at 13:30

jiren

284

In my book a problem is given as follows:

Show that the equation $tan x- x=6$ has one and only one root in the interval $(frac-pi2,fracpi2 )$.

This exercise problem is given after discussing rolle's theorem and mean value theorem so I guess it is an application of these theorems.

calculus real-analysis derivatives

share|cite|improve this question

edited Jul 25 at 13:36

Parcly Taxel

33.5k136588

asked Jul 25 at 13:30

jiren

284

share|cite|improve this question

share|cite|improve this question

edited Jul 25 at 13:36

Parcly Taxel

33.5k136588

edited Jul 25 at 13:36

Parcly Taxel

33.5k136588

edited Jul 25 at 13:36

Parcly Taxel

33.5k136588

33.5k136588

asked Jul 25 at 13:30

jiren

284

asked Jul 25 at 13:30

jiren

284

asked Jul 25 at 13:30

jiren

284

284

closed as off-topic by John Ma, Jendrik Stelzner, John B, Shailesh, Carl Mummert Aug 8 at 1:15

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – John Ma, Jendrik Stelzner, John B, Shailesh, Carl Mummert

If this question can be reworded to fit the rules in the help center, please edit the question.

closed as off-topic by John Ma, Jendrik Stelzner, John B, Shailesh, Carl Mummert Aug 8 at 1:15

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – John Ma, Jendrik Stelzner, John B, Shailesh, Carl Mummert

If this question can be reworded to fit the rules in the help center, please edit the question.

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Yes, you're right about using the mean value theorem! Now suppose there are two such points where $f(x) = tan x - x = 6$. Can you use the MVT to arrive at a contradiction?
  – Cataline
  Jul 25 at 13:35
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Could you please include a more specific source - which book and problem are you looking at?
  – Carl Mummert
  Aug 8 at 1:15
  

  
  
  
  

add a comment | 

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Yes, you're right about using the mean value theorem! Now suppose there are two such points where $f(x) = tan x - x = 6$. Can you use the MVT to arrive at a contradiction?
  – Cataline
  Jul 25 at 13:35
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Could you please include a more specific source - which book and problem are you looking at?
  – Carl Mummert
  Aug 8 at 1:15
  

  
  
  
  

Yes, you're right about using the mean value theorem! Now suppose there are two such points where $f(x) = tan x - x = 6$. Can you use the MVT to arrive at a contradiction?
– Cataline
Jul 25 at 13:35

Yes, you're right about using the mean value theorem! Now suppose there are two such points where $f(x) = tan x - x = 6$. Can you use the MVT to arrive at a contradiction?
– Cataline
Jul 25 at 13:35

Could you please include a more specific source - which book and problem are you looking at?
– Carl Mummert
Aug 8 at 1:15

Could you please include a more specific source - which book and problem are you looking at?
– Carl Mummert
Aug 8 at 1:15

add a comment | 

3 Answers
3

active

oldest

votes

up vote
4
down vote



accepted

Here is a proof using the increasing function theorem, which can be proved as a corollary to the mean value theorem (see comments).

$tan x-x$ is increasing on the given interval with a stationary point at 0 (as can be verified by differentiation). It has limits of $pminfty$ at $pmpi/2$ respectively and is continuous, so its range is the entire real line. Therefore there is only one solution for $tan x-x=6$ (or any other constant replacing 6 here).

share|cite|improve this answer

edited Jul 25 at 13:59

answered Jul 25 at 13:35

Parcly Taxel

33.5k136588

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Isn't the Increasing Function Theorem usually proved as a corollary to MVT? (See the bottom of page 2 www2.clarku.edu/~djoyce/ma120/meanvalue.pdf for example.)
  – Barry Cipra
  Jul 25 at 13:58
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @BarryCipra I have acknowledged this dependency. I have never heard the name "increasing function theorem" - it was never used in my calculus class (MA1102R, National University of Singapore).
  – Parcly Taxel
  Jul 25 at 13:59
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Excellent. I'll leave my comment for the link, unless you want to edit it into the answer, in which case we can delete some clutter. (But first to add to the clutter: You say that IFT can be proved as a corollary to MVT. Is there a proof that doesn't amount to first proving MVT?)
  – Barry Cipra
  Jul 25 at 14:04
  

  
  
  
  

add a comment | 

up vote
2
down vote

HINT

Let consider

• $h(x)=tan x-x-6 implies h'(x)=frac1cos ^2x-1>1$

is continuous and strictly increasing then observe that $h(0)=-6$ and $h(x)to +infty$ as $xto fracpi2^-$ and refer to IVT.

share|cite|improve this answer

answered Jul 25 at 13:36

gimusi

65k73583

add a comment | 

up vote
1
down vote

Hint: Define $$f(x)=tan(x)-x-6$$ then we get
$$f'(x)=frac1cos^2(x)-1$$. Can you proceed?

share|cite|improve this answer

answered Jul 25 at 13:35

Dr. Sonnhard Graubner

66.7k32659

add a comment | 

3 Answers
3

active

oldest

votes

3 Answers
3

active

oldest

votes

active

oldest

votes

active

oldest

votes

up vote
4
down vote



accepted

Here is a proof using the increasing function theorem, which can be proved as a corollary to the mean value theorem (see comments).

$tan x-x$ is increasing on the given interval with a stationary point at 0 (as can be verified by differentiation). It has limits of $pminfty$ at $pmpi/2$ respectively and is continuous, so its range is the entire real line. Therefore there is only one solution for $tan x-x=6$ (or any other constant replacing 6 here).

share|cite|improve this answer

edited Jul 25 at 13:59

answered Jul 25 at 13:35

Parcly Taxel

33.5k136588

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Isn't the Increasing Function Theorem usually proved as a corollary to MVT? (See the bottom of page 2 www2.clarku.edu/~djoyce/ma120/meanvalue.pdf for example.)
  – Barry Cipra
  Jul 25 at 13:58
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @BarryCipra I have acknowledged this dependency. I have never heard the name "increasing function theorem" - it was never used in my calculus class (MA1102R, National University of Singapore).
  – Parcly Taxel
  Jul 25 at 13:59
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Excellent. I'll leave my comment for the link, unless you want to edit it into the answer, in which case we can delete some clutter. (But first to add to the clutter: You say that IFT can be proved as a corollary to MVT. Is there a proof that doesn't amount to first proving MVT?)
  – Barry Cipra
  Jul 25 at 14:04
  

  
  
  
  

add a comment | 

up vote
4
down vote



accepted

Here is a proof using the increasing function theorem, which can be proved as a corollary to the mean value theorem (see comments).

$tan x-x$ is increasing on the given interval with a stationary point at 0 (as can be verified by differentiation). It has limits of $pminfty$ at $pmpi/2$ respectively and is continuous, so its range is the entire real line. Therefore there is only one solution for $tan x-x=6$ (or any other constant replacing 6 here).

share|cite|improve this answer

edited Jul 25 at 13:59

answered Jul 25 at 13:35

Parcly Taxel

33.5k136588

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Isn't the Increasing Function Theorem usually proved as a corollary to MVT? (See the bottom of page 2 www2.clarku.edu/~djoyce/ma120/meanvalue.pdf for example.)
  – Barry Cipra
  Jul 25 at 13:58
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @BarryCipra I have acknowledged this dependency. I have never heard the name "increasing function theorem" - it was never used in my calculus class (MA1102R, National University of Singapore).
  – Parcly Taxel
  Jul 25 at 13:59
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Excellent. I'll leave my comment for the link, unless you want to edit it into the answer, in which case we can delete some clutter. (But first to add to the clutter: You say that IFT can be proved as a corollary to MVT. Is there a proof that doesn't amount to first proving MVT?)
  – Barry Cipra
  Jul 25 at 14:04
  

  
  
  
  

add a comment | 

up vote
4
down vote



accepted

up vote
4
down vote



accepted

Here is a proof using the increasing function theorem, which can be proved as a corollary to the mean value theorem (see comments).

$tan x-x$ is increasing on the given interval with a stationary point at 0 (as can be verified by differentiation). It has limits of $pminfty$ at $pmpi/2$ respectively and is continuous, so its range is the entire real line. Therefore there is only one solution for $tan x-x=6$ (or any other constant replacing 6 here).

share|cite|improve this answer

edited Jul 25 at 13:59

answered Jul 25 at 13:35

Parcly Taxel

33.5k136588

Here is a proof using the increasing function theorem, which can be proved as a corollary to the mean value theorem (see comments).

$tan x-x$ is increasing on the given interval with a stationary point at 0 (as can be verified by differentiation). It has limits of $pminfty$ at $pmpi/2$ respectively and is continuous, so its range is the entire real line. Therefore there is only one solution for $tan x-x=6$ (or any other constant replacing 6 here).

share|cite|improve this answer

edited Jul 25 at 13:59

answered Jul 25 at 13:35

Parcly Taxel

33.5k136588

share|cite|improve this answer

share|cite|improve this answer

edited Jul 25 at 13:59

edited Jul 25 at 13:59

edited Jul 25 at 13:59

answered Jul 25 at 13:35

Parcly Taxel

33.5k136588

answered Jul 25 at 13:35

Parcly Taxel

33.5k136588

answered Jul 25 at 13:35

Parcly Taxel

33.5k136588

33.5k136588

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Isn't the Increasing Function Theorem usually proved as a corollary to MVT? (See the bottom of page 2 www2.clarku.edu/~djoyce/ma120/meanvalue.pdf for example.)
  – Barry Cipra
  Jul 25 at 13:58
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @BarryCipra I have acknowledged this dependency. I have never heard the name "increasing function theorem" - it was never used in my calculus class (MA1102R, National University of Singapore).
  – Parcly Taxel
  Jul 25 at 13:59
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Excellent. I'll leave my comment for the link, unless you want to edit it into the answer, in which case we can delete some clutter. (But first to add to the clutter: You say that IFT can be proved as a corollary to MVT. Is there a proof that doesn't amount to first proving MVT?)
  – Barry Cipra
  Jul 25 at 14:04
  

  
  
  
  

add a comment | 

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Isn't the Increasing Function Theorem usually proved as a corollary to MVT? (See the bottom of page 2 www2.clarku.edu/~djoyce/ma120/meanvalue.pdf for example.)
  – Barry Cipra
  Jul 25 at 13:58
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @BarryCipra I have acknowledged this dependency. I have never heard the name "increasing function theorem" - it was never used in my calculus class (MA1102R, National University of Singapore).
  – Parcly Taxel
  Jul 25 at 13:59
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Excellent. I'll leave my comment for the link, unless you want to edit it into the answer, in which case we can delete some clutter. (But first to add to the clutter: You say that IFT can be proved as a corollary to MVT. Is there a proof that doesn't amount to first proving MVT?)
  – Barry Cipra
  Jul 25 at 14:04
  

  
  
  
  

Isn't the Increasing Function Theorem usually proved as a corollary to MVT? (See the bottom of page 2 www2.clarku.edu/~djoyce/ma120/meanvalue.pdf for example.)
– Barry Cipra
Jul 25 at 13:58

Isn't the Increasing Function Theorem usually proved as a corollary to MVT? (See the bottom of page 2 www2.clarku.edu/~djoyce/ma120/meanvalue.pdf for example.)
– Barry Cipra
Jul 25 at 13:58

@BarryCipra I have acknowledged this dependency. I have never heard the name "increasing function theorem" - it was never used in my calculus class (MA1102R, National University of Singapore).
– Parcly Taxel
Jul 25 at 13:59

@BarryCipra I have acknowledged this dependency. I have never heard the name "increasing function theorem" - it was never used in my calculus class (MA1102R, National University of Singapore).
– Parcly Taxel
Jul 25 at 13:59

Excellent. I'll leave my comment for the link, unless you want to edit it into the answer, in which case we can delete some clutter. (But first to add to the clutter: You say that IFT can be proved as a corollary to MVT. Is there a proof that doesn't amount to first proving MVT?)
– Barry Cipra
Jul 25 at 14:04

Excellent. I'll leave my comment for the link, unless you want to edit it into the answer, in which case we can delete some clutter. (But first to add to the clutter: You say that IFT can be proved as a corollary to MVT. Is there a proof that doesn't amount to first proving MVT?)
– Barry Cipra
Jul 25 at 14:04

add a comment | 

up vote
2
down vote

HINT

Let consider

• $h(x)=tan x-x-6 implies h'(x)=frac1cos ^2x-1>1$

is continuous and strictly increasing then observe that $h(0)=-6$ and $h(x)to +infty$ as $xto fracpi2^-$ and refer to IVT.

share|cite|improve this answer

answered Jul 25 at 13:36

gimusi

65k73583

add a comment | 

up vote
2
down vote

HINT

Let consider

• $h(x)=tan x-x-6 implies h'(x)=frac1cos ^2x-1>1$

is continuous and strictly increasing then observe that $h(0)=-6$ and $h(x)to +infty$ as $xto fracpi2^-$ and refer to IVT.

share|cite|improve this answer

answered Jul 25 at 13:36

gimusi

65k73583

add a comment | 

up vote
2
down vote

up vote
2
down vote

HINT

Let consider

• $h(x)=tan x-x-6 implies h'(x)=frac1cos ^2x-1>1$

is continuous and strictly increasing then observe that $h(0)=-6$ and $h(x)to +infty$ as $xto fracpi2^-$ and refer to IVT.

share|cite|improve this answer

answered Jul 25 at 13:36

gimusi

65k73583

HINT

Let consider

• $h(x)=tan x-x-6 implies h'(x)=frac1cos ^2x-1>1$

is continuous and strictly increasing then observe that $h(0)=-6$ and $h(x)to +infty$ as $xto fracpi2^-$ and refer to IVT.

share|cite|improve this answer

answered Jul 25 at 13:36

gimusi

65k73583

share|cite|improve this answer

share|cite|improve this answer

answered Jul 25 at 13:36

gimusi

65k73583

answered Jul 25 at 13:36

gimusi

65k73583

answered Jul 25 at 13:36

gimusi

65k73583

65k73583

add a comment | 

add a comment | 

up vote
1
down vote

Hint: Define $$f(x)=tan(x)-x-6$$ then we get
$$f'(x)=frac1cos^2(x)-1$$. Can you proceed?

share|cite|improve this answer

answered Jul 25 at 13:35

Dr. Sonnhard Graubner

66.7k32659

add a comment | 

up vote
1
down vote

Hint: Define $$f(x)=tan(x)-x-6$$ then we get
$$f'(x)=frac1cos^2(x)-1$$. Can you proceed?

share|cite|improve this answer

answered Jul 25 at 13:35

Dr. Sonnhard Graubner

66.7k32659

add a comment | 

up vote
1
down vote

up vote
1
down vote

Hint: Define $$f(x)=tan(x)-x-6$$ then we get
$$f'(x)=frac1cos^2(x)-1$$. Can you proceed?

share|cite|improve this answer

answered Jul 25 at 13:35

Dr. Sonnhard Graubner

66.7k32659

Hint: Define $$f(x)=tan(x)-x-6$$ then we get
$$f'(x)=frac1cos^2(x)-1$$. Can you proceed?

share|cite|improve this answer

answered Jul 25 at 13:35

Dr. Sonnhard Graubner

66.7k32659

share|cite|improve this answer

share|cite|improve this answer

answered Jul 25 at 13:35

Dr. Sonnhard Graubner

66.7k32659

answered Jul 25 at 13:35

Dr. Sonnhard Graubner

66.7k32659

answered Jul 25 at 13:35

Dr. Sonnhard Graubner

66.7k32659

66.7k32659

add a comment | 

add a comment |