How to determine local extrema for $f(x) = xcdot sin(x) ^ {sin(x)}$
up vote
2
down vote
favorite
I need to find the local extrema points of the following function:
$f(x) = xcdotsin(x) ^ {sin(x)}$
I was already able to derive to this function:
$f'(x) = x (ln(sin(x))+1)cos(x)sin(x)^{sin(x)}+sin(x)^ {sin(x)}$
calculus derivatives trigonometry maxima-minima
edited Nov 23 at 20:02
David G. Stork
9,54721232
asked Nov 23 at 19:40
Phins
132
|
show 5 more comments
up vote
2
down vote
favorite
I need to find the local extrema points of the following function:
$f(x) = xcdotsin(x) ^ {sin(x)}$
I was already able to derive to this function:
$f'(x) = x (ln(sin(x))+1)cos(x)sin(x)^{sin(x)}+sin(x)^ {sin(x)}$
calculus derivatives trigonometry maxima-minima
edited Nov 23 at 20:02
David G. Stork
9,54721232
asked Nov 23 at 19:40
Phins
132
find $x$ where $f'(x)$ is equal to $0$. Then assess the sign change while crossing the roots
– Makina
Nov 23 at 19:45
This function is defined in $bigcuplimits_{ninmathbb Z}[2npi,2npi+pi]$.
– Federico
Nov 23 at 19:50
You first have to take into considerations the extreme points of the intervals
– Federico
Nov 23 at 19:51
Then your equation $f'=0$ inside these intervals is highly non-linear and trascendental...
– Federico
Nov 23 at 19:52
I don't think much can be said, except numerically
– Federico
Nov 23 at 19:53
|
show 5 more comments
up vote
2
down vote
favorite
up vote
2
down vote
favorite
I need to find the local extrema points of the following function:
$f(x) = xcdotsin(x) ^ {sin(x)}$
I was already able to derive to this function:
$f'(x) = x (ln(sin(x))+1)cos(x)sin(x)^{sin(x)}+sin(x)^ {sin(x)}$
calculus derivatives trigonometry maxima-minima
edited Nov 23 at 20:02
David G. Stork
9,54721232
asked Nov 23 at 19:40
Phins
132
I need to find the local extrema points of the following function:
$f(x) = xcdotsin(x) ^ {sin(x)}$
I was already able to derive to this function:
$f'(x) = x (ln(sin(x))+1)cos(x)sin(x)^{sin(x)}+sin(x)^ {sin(x)}$
calculus derivatives trigonometry maxima-minima
calculus derivatives trigonometry maxima-minima
edited Nov 23 at 20:02
David G. Stork
9,54721232
asked Nov 23 at 19:40
Phins
132
edited Nov 23 at 20:02
David G. Stork
9,54721232
asked Nov 23 at 19:40
Phins
132
edited Nov 23 at 20:02
David G. Stork
9,54721232
edited Nov 23 at 20:02
David G. Stork
9,54721232
edited Nov 23 at 20:02
David G. Stork
9,54721232
9,54721232
asked Nov 23 at 19:40
Phins
132
asked Nov 23 at 19:40
Phins
132
asked Nov 23 at 19:40
Phins
132
132
find $x$ where $f'(x)$ is equal to $0$. Then assess the sign change while crossing the roots
– Makina
Nov 23 at 19:45
This function is defined in $bigcuplimits_{ninmathbb Z}[2npi,2npi+pi]$.
– Federico
Nov 23 at 19:50
You first have to take into considerations the extreme points of the intervals
– Federico
Nov 23 at 19:51
Then your equation $f'=0$ inside these intervals is highly non-linear and trascendental...
– Federico
Nov 23 at 19:52
I don't think much can be said, except numerically
– Federico
Nov 23 at 19:53
|
show 5 more comments
find $x$ where $f'(x)$ is equal to $0$. Then assess the sign change while crossing the roots
– Makina
Nov 23 at 19:45
This function is defined in $bigcuplimits_{ninmathbb Z}[2npi,2npi+pi]$.
– Federico
Nov 23 at 19:50
You first have to take into considerations the extreme points of the intervals
– Federico
Nov 23 at 19:51
Then your equation $f'=0$ inside these intervals is highly non-linear and trascendental...
– Federico
Nov 23 at 19:52
I don't think much can be said, except numerically
– Federico
Nov 23 at 19:53
find $x$ where $f'(x)$ is equal to $0$. Then assess the sign change while crossing the roots
– Makina
Nov 23 at 19:45
find $x$ where $f'(x)$ is equal to $0$. Then assess the sign change while crossing the roots
– Makina
Nov 23 at 19:45
This function is defined in $bigcuplimits_{ninmathbb Z}[2npi,2npi+pi]$.
– Federico
Nov 23 at 19:50
This function is defined in $bigcuplimits_{ninmathbb Z}[2npi,2npi+pi]$.
– Federico
Nov 23 at 19:50
You first have to take into considerations the extreme points of the intervals
– Federico
Nov 23 at 19:51
You first have to take into considerations the extreme points of the intervals
– Federico
Nov 23 at 19:51
Then your equation $f'=0$ inside these intervals is highly non-linear and trascendental...
– Federico
Nov 23 at 19:52
Then your equation $f'=0$ inside these intervals is highly non-linear and trascendental...
– Federico
Nov 23 at 19:52
I don't think much can be said, except numerically
– Federico
Nov 23 at 19:53
I don't think much can be said, except numerically
– Federico
Nov 23 at 19:53
|
show 5 more comments
1 Answer
1
active
oldest
votes
up vote
2
down vote
accepted
The domain of $f$ is where $sin x>0$ i.e. $$bigcup_{nin Bbb Z}(2npi ,2npi +pi)$$ on this domain by equaling the derivative to zero we obtain $$sin x^{sin x}=0\text{or}\ x(1+ln sin x)cos x+1=0$$where $sin x^{sin x}=0$ is always impossible and the second equation can only be solved numerically. Here is a sketch of the function
answered Nov 23 at 20:19
Mostafa Ayaz
13.6k3836
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
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
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3010753%2fhow-to-determine-local-extrema-for-fx-x-cdot-sinx-sinx%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
accepted
The domain of $f$ is where $sin x>0$ i.e. $$bigcup_{nin Bbb Z}(2npi ,2npi +pi)$$ on this domain by equaling the derivative to zero we obtain $$sin x^{sin x}=0\text{or}\ x(1+ln sin x)cos x+1=0$$where $sin x^{sin x}=0$ is always impossible and the second equation can only be solved numerically. Here is a sketch of the function
answered Nov 23 at 20:19
Mostafa Ayaz
13.6k3836
add a comment |
up vote
2
down vote
accepted
The domain of $f$ is where $sin x>0$ i.e. $$bigcup_{nin Bbb Z}(2npi ,2npi +pi)$$ on this domain by equaling the derivative to zero we obtain $$sin x^{sin x}=0\text{or}\ x(1+ln sin x)cos x+1=0$$where $sin x^{sin x}=0$ is always impossible and the second equation can only be solved numerically. Here is a sketch of the function
answered Nov 23 at 20:19
Mostafa Ayaz
13.6k3836
add a comment |
up vote
2
down vote
accepted
up vote
2
down vote
accepted
The domain of $f$ is where $sin x>0$ i.e. $$bigcup_{nin Bbb Z}(2npi ,2npi +pi)$$ on this domain by equaling the derivative to zero we obtain $$sin x^{sin x}=0\text{or}\ x(1+ln sin x)cos x+1=0$$where $sin x^{sin x}=0$ is always impossible and the second equation can only be solved numerically. Here is a sketch of the function
answered Nov 23 at 20:19
Mostafa Ayaz
13.6k3836
The domain of $f$ is where $sin x>0$ i.e. $$bigcup_{nin Bbb Z}(2npi ,2npi +pi)$$ on this domain by equaling the derivative to zero we obtain $$sin x^{sin x}=0\text{or}\ x(1+ln sin x)cos x+1=0$$where $sin x^{sin x}=0$ is always impossible and the second equation can only be solved numerically. Here is a sketch of the function
answered Nov 23 at 20:19
Mostafa Ayaz
13.6k3836
answered Nov 23 at 20:19
Mostafa Ayaz
13.6k3836
answered Nov 23 at 20:19
Mostafa Ayaz
13.6k3836
answered Nov 23 at 20:19
Mostafa Ayaz
13.6k3836
13.6k3836
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
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
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3010753%2fhow-to-determine-local-extrema-for-fx-x-cdot-sinx-sinx%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
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
Required, but never shown
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
Required, but never shown
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
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
find $x$ where $f'(x)$ is equal to $0$. Then assess the sign change while crossing the roots
– Makina
Nov 23 at 19:45
This function is defined in $bigcuplimits_{ninmathbb Z}[2npi,2npi+pi]$.
– Federico
Nov 23 at 19:50
You first have to take into considerations the extreme points of the intervals
– Federico
Nov 23 at 19:51
Then your equation $f'=0$ inside these intervals is highly non-linear and trascendental...
– Federico
Nov 23 at 19:52
I don't think much can be said, except numerically
– Federico
Nov 23 at 19:53