Mixing
What are the applications of the extreme value theorem?
Clash Royale CLAN TAG#URR8PPP
up vote
1
down vote
favorite
I recently started studying about global maxima and minima.
The extreme value theorem proves the existence of global maxima and minima for a continuous function in a closed interval.This theorem is actually, pretty much intuitive and looks like a basic property of a continuous function.
Has it got any applications other than just proving existence of global maxima or minima?
If you take a look at Lagrange's mean value theorem we can use it to prove inequalities like $|cos(a)-cos (b)|<|a-b|$.
So, I'm just looking for some cool stuff that you can do using the extreme value theorem.
calculus functions soft-question
edited Jul 28 at 5:53
高田航
1,116318
asked Jul 28 at 5:15
Banchin
112
add a comment |Â
up vote
1
down vote
favorite
I recently started studying about global maxima and minima.
The extreme value theorem proves the existence of global maxima and minima for a continuous function in a closed interval.This theorem is actually, pretty much intuitive and looks like a basic property of a continuous function.
Has it got any applications other than just proving existence of global maxima or minima?
If you take a look at Lagrange's mean value theorem we can use it to prove inequalities like $|cos(a)-cos (b)|<|a-b|$.
So, I'm just looking for some cool stuff that you can do using the extreme value theorem.
calculus functions soft-question
edited Jul 28 at 5:53
高田航
1,116318
asked Jul 28 at 5:15
Banchin
112
It's used in the proof of many other theorems.
– Dole
Jul 28 at 5:24
By extreme value theorem you mean Weierstrass theorem? (i.e. a real continuous function on a compact set reaches is maximum and minimum?)
– Bob
Jul 28 at 6:04
Not exactly applications, but some perks and quirks of the extreme value theorem are: it is a consequece of a far more general (and simpler) fact of topology that the image of a compact set trough a continuous function is again a compact set and the fact that a compact set on the real line is closed and bounded (not very simple to prove) and that it may fail to hold depending on how you construct the real numbers (using intuitionistic logic instead of classical logic).
– Henrique Augusto Souza
Jul 28 at 6:53
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I recently started studying about global maxima and minima.
The extreme value theorem proves the existence of global maxima and minima for a continuous function in a closed interval.This theorem is actually, pretty much intuitive and looks like a basic property of a continuous function.
Has it got any applications other than just proving existence of global maxima or minima?
If you take a look at Lagrange's mean value theorem we can use it to prove inequalities like $|cos(a)-cos (b)|<|a-b|$.
So, I'm just looking for some cool stuff that you can do using the extreme value theorem.
calculus functions soft-question
edited Jul 28 at 5:53
高田航
1,116318
asked Jul 28 at 5:15
Banchin
112
I recently started studying about global maxima and minima.
The extreme value theorem proves the existence of global maxima and minima for a continuous function in a closed interval.This theorem is actually, pretty much intuitive and looks like a basic property of a continuous function.
Has it got any applications other than just proving existence of global maxima or minima?
If you take a look at Lagrange's mean value theorem we can use it to prove inequalities like $|cos(a)-cos (b)|<|a-b|$.
So, I'm just looking for some cool stuff that you can do using the extreme value theorem.
calculus functions soft-question
edited Jul 28 at 5:53
高田航
1,116318
asked Jul 28 at 5:15
Banchin
112
edited Jul 28 at 5:53
高田航
1,116318
edited Jul 28 at 5:53
高田航
1,116318
edited Jul 28 at 5:53
高田航
1,116318
1,116318
asked Jul 28 at 5:15
Banchin
112
asked Jul 28 at 5:15
Banchin
112
asked Jul 28 at 5:15
Banchin
112
112
It's used in the proof of many other theorems.
– Dole
Jul 28 at 5:24
By extreme value theorem you mean Weierstrass theorem? (i.e. a real continuous function on a compact set reaches is maximum and minimum?)
– Bob
Jul 28 at 6:04
Not exactly applications, but some perks and quirks of the extreme value theorem are: it is a consequece of a far more general (and simpler) fact of topology that the image of a compact set trough a continuous function is again a compact set and the fact that a compact set on the real line is closed and bounded (not very simple to prove) and that it may fail to hold depending on how you construct the real numbers (using intuitionistic logic instead of classical logic).
– Henrique Augusto Souza
Jul 28 at 6:53
add a comment |Â
It's used in the proof of many other theorems.
– Dole
Jul 28 at 5:24
By extreme value theorem you mean Weierstrass theorem? (i.e. a real continuous function on a compact set reaches is maximum and minimum?)
– Bob
Jul 28 at 6:04
Not exactly applications, but some perks and quirks of the extreme value theorem are: it is a consequece of a far more general (and simpler) fact of topology that the image of a compact set trough a continuous function is again a compact set and the fact that a compact set on the real line is closed and bounded (not very simple to prove) and that it may fail to hold depending on how you construct the real numbers (using intuitionistic logic instead of classical logic).
– Henrique Augusto Souza
Jul 28 at 6:53
It's used in the proof of many other theorems.
– Dole
Jul 28 at 5:24
It's used in the proof of many other theorems.
– Dole
Jul 28 at 5:24
By extreme value theorem you mean Weierstrass theorem? (i.e. a real continuous function on a compact set reaches is maximum and minimum?)
– Bob
Jul 28 at 6:04
By extreme value theorem you mean Weierstrass theorem? (i.e. a real continuous function on a compact set reaches is maximum and minimum?)
– Bob
Jul 28 at 6:04
Not exactly applications, but some perks and quirks of the extreme value theorem are: it is a consequece of a far more general (and simpler) fact of topology that the image of a compact set trough a continuous function is again a compact set and the fact that a compact set on the real line is closed and bounded (not very simple to prove) and that it may fail to hold depending on how you construct the real numbers (using intuitionistic logic instead of classical logic).
– Henrique Augusto Souza
Jul 28 at 6:53
Not exactly applications, but some perks and quirks of the extreme value theorem are: it is a consequece of a far more general (and simpler) fact of topology that the image of a compact set trough a continuous function is again a compact set and the fact that a compact set on the real line is closed and bounded (not very simple to prove) and that it may fail to hold depending on how you construct the real numbers (using intuitionistic logic instead of classical logic).
– Henrique Augusto Souza
Jul 28 at 6:53
add a comment |Â
2 Answers
2
active
oldest
votes
up vote
2
down vote
The Extreme Value Theorem sits in the middle of a chain of theorems, where each theorem is proved using the last.
LUB Property $rightarrow$ Monotone bounded convergence $rightarrow$ Bolzano-Weierstrass $rightarrow$ EVT $rightarrow$ IVT $rightarrow$ Rolle's Theorem $rightarrow$ MVT, Cauchy's MVT $rightarrow$ Integral MVT
Each theorem in the chain is useful by itself, but the main use of EVT is to act as a stepping stone in this chain.
answered Jul 28 at 6:09
Display name
623211
add a comment |Â
up vote
0
down vote
Well the extreme value theorem is very nice for bounding how fast your functions can grow. Given an arbitrary $f in (a,b)$ that is continuous, we may not guarantee that it's bounded. Indeed, take $f = frac1x$, then it's continuous on $(0,1)$, but not bounded.
Here's one example of how useful this theorem is: Let $f$ be continuously differentiable on $[a,b]$, then $f$ is Lipschitz continuous. Then by EVT, $f'$ is bounded by say $M$
Indeed, by MVT, for any $x,yin (a,b)$, we can find a $zeta in (a,b)$ such that beginequation |f(x)-f(y)| = |f'(zeta)||x-y| leq M|x-y|endequation
so we see $f$ is not just uniformly continuous, but Lipschitz.
Another useful trick for it is to bound how big and small your integral can be given that $f$ is continuous on $[a,b]$. Say $m leq f leq M$, where $f:[0,1] rightarrow mathbbR$. Then we have beginequation int_0^1 m leq int_0^1 f leq int_0^1 Mendequation Then we see the end points by EVT are points that $f$ attains, so by IVT, there is some $zeta$ such that $f(zeta) = int_0^1 f$
In general, it is a very powerful bounding tool that can coupled with IVT to produce nice results.
answered Jul 28 at 17:42
Raymond Chu
1,02219
add a comment |Â
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
The Extreme Value Theorem sits in the middle of a chain of theorems, where each theorem is proved using the last.
LUB Property $rightarrow$ Monotone bounded convergence $rightarrow$ Bolzano-Weierstrass $rightarrow$ EVT $rightarrow$ IVT $rightarrow$ Rolle's Theorem $rightarrow$ MVT, Cauchy's MVT $rightarrow$ Integral MVT
Each theorem in the chain is useful by itself, but the main use of EVT is to act as a stepping stone in this chain.
answered Jul 28 at 6:09
Display name
623211
add a comment |Â
up vote
2
down vote
The Extreme Value Theorem sits in the middle of a chain of theorems, where each theorem is proved using the last.
LUB Property $rightarrow$ Monotone bounded convergence $rightarrow$ Bolzano-Weierstrass $rightarrow$ EVT $rightarrow$ IVT $rightarrow$ Rolle's Theorem $rightarrow$ MVT, Cauchy's MVT $rightarrow$ Integral MVT
Each theorem in the chain is useful by itself, but the main use of EVT is to act as a stepping stone in this chain.
answered Jul 28 at 6:09
Display name
623211
add a comment |Â
up vote
2
down vote
up vote
2
down vote
The Extreme Value Theorem sits in the middle of a chain of theorems, where each theorem is proved using the last.
LUB Property $rightarrow$ Monotone bounded convergence $rightarrow$ Bolzano-Weierstrass $rightarrow$ EVT $rightarrow$ IVT $rightarrow$ Rolle's Theorem $rightarrow$ MVT, Cauchy's MVT $rightarrow$ Integral MVT
Each theorem in the chain is useful by itself, but the main use of EVT is to act as a stepping stone in this chain.
answered Jul 28 at 6:09
Display name
623211
The Extreme Value Theorem sits in the middle of a chain of theorems, where each theorem is proved using the last.
LUB Property $rightarrow$ Monotone bounded convergence $rightarrow$ Bolzano-Weierstrass $rightarrow$ EVT $rightarrow$ IVT $rightarrow$ Rolle's Theorem $rightarrow$ MVT, Cauchy's MVT $rightarrow$ Integral MVT
Each theorem in the chain is useful by itself, but the main use of EVT is to act as a stepping stone in this chain.
answered Jul 28 at 6:09
Display name
623211
answered Jul 28 at 6:09
Display name
623211
answered Jul 28 at 6:09
Display name
623211
answered Jul 28 at 6:09
Display name
623211
623211
add a comment |Â
add a comment |Â
up vote
0
down vote
Well the extreme value theorem is very nice for bounding how fast your functions can grow. Given an arbitrary $f in (a,b)$ that is continuous, we may not guarantee that it's bounded. Indeed, take $f = frac1x$, then it's continuous on $(0,1)$, but not bounded.
Here's one example of how useful this theorem is: Let $f$ be continuously differentiable on $[a,b]$, then $f$ is Lipschitz continuous. Then by EVT, $f'$ is bounded by say $M$
Indeed, by MVT, for any $x,yin (a,b)$, we can find a $zeta in (a,b)$ such that beginequation |f(x)-f(y)| = |f'(zeta)||x-y| leq M|x-y|endequation
so we see $f$ is not just uniformly continuous, but Lipschitz.
Another useful trick for it is to bound how big and small your integral can be given that $f$ is continuous on $[a,b]$. Say $m leq f leq M$, where $f:[0,1] rightarrow mathbbR$. Then we have beginequation int_0^1 m leq int_0^1 f leq int_0^1 Mendequation Then we see the end points by EVT are points that $f$ attains, so by IVT, there is some $zeta$ such that $f(zeta) = int_0^1 f$
In general, it is a very powerful bounding tool that can coupled with IVT to produce nice results.
answered Jul 28 at 17:42
Raymond Chu
1,02219
add a comment |Â
up vote
0
down vote
Well the extreme value theorem is very nice for bounding how fast your functions can grow. Given an arbitrary $f in (a,b)$ that is continuous, we may not guarantee that it's bounded. Indeed, take $f = frac1x$, then it's continuous on $(0,1)$, but not bounded.
Here's one example of how useful this theorem is: Let $f$ be continuously differentiable on $[a,b]$, then $f$ is Lipschitz continuous. Then by EVT, $f'$ is bounded by say $M$
Indeed, by MVT, for any $x,yin (a,b)$, we can find a $zeta in (a,b)$ such that beginequation |f(x)-f(y)| = |f'(zeta)||x-y| leq M|x-y|endequation
so we see $f$ is not just uniformly continuous, but Lipschitz.
Another useful trick for it is to bound how big and small your integral can be given that $f$ is continuous on $[a,b]$. Say $m leq f leq M$, where $f:[0,1] rightarrow mathbbR$. Then we have beginequation int_0^1 m leq int_0^1 f leq int_0^1 Mendequation Then we see the end points by EVT are points that $f$ attains, so by IVT, there is some $zeta$ such that $f(zeta) = int_0^1 f$
In general, it is a very powerful bounding tool that can coupled with IVT to produce nice results.
answered Jul 28 at 17:42
Raymond Chu
1,02219
add a comment |Â
up vote
0
down vote
up vote
0
down vote
Well the extreme value theorem is very nice for bounding how fast your functions can grow. Given an arbitrary $f in (a,b)$ that is continuous, we may not guarantee that it's bounded. Indeed, take $f = frac1x$, then it's continuous on $(0,1)$, but not bounded.
Here's one example of how useful this theorem is: Let $f$ be continuously differentiable on $[a,b]$, then $f$ is Lipschitz continuous. Then by EVT, $f'$ is bounded by say $M$
Indeed, by MVT, for any $x,yin (a,b)$, we can find a $zeta in (a,b)$ such that beginequation |f(x)-f(y)| = |f'(zeta)||x-y| leq M|x-y|endequation
so we see $f$ is not just uniformly continuous, but Lipschitz.
Another useful trick for it is to bound how big and small your integral can be given that $f$ is continuous on $[a,b]$. Say $m leq f leq M$, where $f:[0,1] rightarrow mathbbR$. Then we have beginequation int_0^1 m leq int_0^1 f leq int_0^1 Mendequation Then we see the end points by EVT are points that $f$ attains, so by IVT, there is some $zeta$ such that $f(zeta) = int_0^1 f$
In general, it is a very powerful bounding tool that can coupled with IVT to produce nice results.
answered Jul 28 at 17:42
Raymond Chu
1,02219
Well the extreme value theorem is very nice for bounding how fast your functions can grow. Given an arbitrary $f in (a,b)$ that is continuous, we may not guarantee that it's bounded. Indeed, take $f = frac1x$, then it's continuous on $(0,1)$, but not bounded.
Here's one example of how useful this theorem is: Let $f$ be continuously differentiable on $[a,b]$, then $f$ is Lipschitz continuous. Then by EVT, $f'$ is bounded by say $M$
Indeed, by MVT, for any $x,yin (a,b)$, we can find a $zeta in (a,b)$ such that beginequation |f(x)-f(y)| = |f'(zeta)||x-y| leq M|x-y|endequation
so we see $f$ is not just uniformly continuous, but Lipschitz.
Another useful trick for it is to bound how big and small your integral can be given that $f$ is continuous on $[a,b]$. Say $m leq f leq M$, where $f:[0,1] rightarrow mathbbR$. Then we have beginequation int_0^1 m leq int_0^1 f leq int_0^1 Mendequation Then we see the end points by EVT are points that $f$ attains, so by IVT, there is some $zeta$ such that $f(zeta) = int_0^1 f$
In general, it is a very powerful bounding tool that can coupled with IVT to produce nice results.
answered Jul 28 at 17:42
Raymond Chu
1,02219
answered Jul 28 at 17:42
Raymond Chu
1,02219
answered Jul 28 at 17:42
Raymond Chu
1,02219
answered Jul 28 at 17:42
Raymond Chu
1,02219
1,02219
add a comment |Â
add a comment |Â
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2865000%2fwhat-are-the-applications-of-the-extreme-value-theorem%23new-answer', 'question_page');
);
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
It's used in the proof of many other theorems.
– Dole
Jul 28 at 5:24
By extreme value theorem you mean Weierstrass theorem? (i.e. a real continuous function on a compact set reaches is maximum and minimum?)
– Bob
Jul 28 at 6:04
Not exactly applications, but some perks and quirks of the extreme value theorem are: it is a consequece of a far more general (and simpler) fact of topology that the image of a compact set trough a continuous function is again a compact set and the fact that a compact set on the real line is closed and bounded (not very simple to prove) and that it may fail to hold depending on how you construct the real numbers (using intuitionistic logic instead of classical logic).
– Henrique Augusto Souza
Jul 28 at 6:53