maxima/minima $h(v,w,x,y) := 6v^2-12v+arctan(w)- frac{1}{2}w+exp(x^2)+x^2+y^2+frac{1}{4}xy$
Let $h: {(v,w,x,y) in mathbb{R}^4 : w <0 } to mathbb{R}$ with $h(v,w,x,y) := 6v^2-12v+arctan(w)- frac{1}{2}w+exp(x^2)+x^2+y^2+frac{1}{4}xy$
How can one find the criticial points, i.e. the local/global maxima and minima and saddle points of this function?
I know that a local maxima/minima $x_E$ of a function is its Zero of its derivative $f'(x_E) = 0$.
If $f''(x_E) > 0$ then there's a local minimum.
If $f''(x_E) < 0$ then there's a local maximum.
And if $f''(x_E) = 0$ we can't tell anything.
I don't know how to derivate the function twice, because of the condition that $w <0$ and how I should proceed afterwards.
analysis derivatives maxima-minima
asked Nov 30 '18 at 1:53
Math DummyMath Dummy
276
add a comment |
Let $h: {(v,w,x,y) in mathbb{R}^4 : w <0 } to mathbb{R}$ with $h(v,w,x,y) := 6v^2-12v+arctan(w)- frac{1}{2}w+exp(x^2)+x^2+y^2+frac{1}{4}xy$
How can one find the criticial points, i.e. the local/global maxima and minima and saddle points of this function?
I know that a local maxima/minima $x_E$ of a function is its Zero of its derivative $f'(x_E) = 0$.
If $f''(x_E) > 0$ then there's a local minimum.
If $f''(x_E) < 0$ then there's a local maximum.
And if $f''(x_E) = 0$ we can't tell anything.
I don't know how to derivate the function twice, because of the condition that $w <0$ and how I should proceed afterwards.
analysis derivatives maxima-minima
asked Nov 30 '18 at 1:53
Math DummyMath Dummy
276
add a comment |
Let $h: {(v,w,x,y) in mathbb{R}^4 : w <0 } to mathbb{R}$ with $h(v,w,x,y) := 6v^2-12v+arctan(w)- frac{1}{2}w+exp(x^2)+x^2+y^2+frac{1}{4}xy$
How can one find the criticial points, i.e. the local/global maxima and minima and saddle points of this function?
I know that a local maxima/minima $x_E$ of a function is its Zero of its derivative $f'(x_E) = 0$.
If $f''(x_E) > 0$ then there's a local minimum.
If $f''(x_E) < 0$ then there's a local maximum.
And if $f''(x_E) = 0$ we can't tell anything.
I don't know how to derivate the function twice, because of the condition that $w <0$ and how I should proceed afterwards.
analysis derivatives maxima-minima
asked Nov 30 '18 at 1:53
Math DummyMath Dummy
276
Let $h: {(v,w,x,y) in mathbb{R}^4 : w <0 } to mathbb{R}$ with $h(v,w,x,y) := 6v^2-12v+arctan(w)- frac{1}{2}w+exp(x^2)+x^2+y^2+frac{1}{4}xy$
How can one find the criticial points, i.e. the local/global maxima and minima and saddle points of this function?
I know that a local maxima/minima $x_E$ of a function is its Zero of its derivative $f'(x_E) = 0$.
If $f''(x_E) > 0$ then there's a local minimum.
If $f''(x_E) < 0$ then there's a local maximum.
And if $f''(x_E) = 0$ we can't tell anything.
I don't know how to derivate the function twice, because of the condition that $w <0$ and how I should proceed afterwards.
analysis derivatives maxima-minima
analysis derivatives maxima-minima
asked Nov 30 '18 at 1:53
Math DummyMath Dummy
276
asked Nov 30 '18 at 1:53
Math DummyMath Dummy
276
asked Nov 30 '18 at 1:53
Math DummyMath Dummy
276
asked Nov 30 '18 at 1:53
Math DummyMath Dummy
276
asked Nov 30 '18 at 1:53
Math DummyMath Dummy
276
276
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
The equilibrium points will be where the gradient $nabla h = (h_v,h_w,h_x,h_y)$ is the $0$ vector. To tell whether or not it is a maximum/minimum/saddle, you have to look at the eigenvalues of the Hessian matrix,
$$
H = begin{pmatrix}
h_{vv} & h_{vw} & h_{vx} & h_{vy} \
h_{wv} & h_{ww} & h_{wx} & h_{wy} \
h_{xv} & h_{xw} & h_{xv} & h_{xy} \
h_{yv} & h_{yw} & h_{yx} & h_{yy}
end{pmatrix}.
$$
By the equality of mixed partials the eigenvalues are real and the equilibrium point will be a minimum if each eigenvalue is positive, a maximum if they are all negative, and a saddle if they have different signs. $0$ eigenvalue is a still an indeterminate case.
You will also have to be careful with your domain. Since it does not contain the hyperplane $w=0$, it is possible that the extrema will lie on this hyperplane but your function cannot reach it in this domain.
answered Nov 30 '18 at 2:17
whpowell96whpowell96
53115
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%2f3019514%2fmaxima-minima-hv-w-x-y-6v2-12varctanw-frac12w-expx2x2y2%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
The equilibrium points will be where the gradient $nabla h = (h_v,h_w,h_x,h_y)$ is the $0$ vector. To tell whether or not it is a maximum/minimum/saddle, you have to look at the eigenvalues of the Hessian matrix,
$$
H = begin{pmatrix}
h_{vv} & h_{vw} & h_{vx} & h_{vy} \
h_{wv} & h_{ww} & h_{wx} & h_{wy} \
h_{xv} & h_{xw} & h_{xv} & h_{xy} \
h_{yv} & h_{yw} & h_{yx} & h_{yy}
end{pmatrix}.
$$
By the equality of mixed partials the eigenvalues are real and the equilibrium point will be a minimum if each eigenvalue is positive, a maximum if they are all negative, and a saddle if they have different signs. $0$ eigenvalue is a still an indeterminate case.
You will also have to be careful with your domain. Since it does not contain the hyperplane $w=0$, it is possible that the extrema will lie on this hyperplane but your function cannot reach it in this domain.
answered Nov 30 '18 at 2:17
whpowell96whpowell96
53115
add a comment |
The equilibrium points will be where the gradient $nabla h = (h_v,h_w,h_x,h_y)$ is the $0$ vector. To tell whether or not it is a maximum/minimum/saddle, you have to look at the eigenvalues of the Hessian matrix,
$$
H = begin{pmatrix}
h_{vv} & h_{vw} & h_{vx} & h_{vy} \
h_{wv} & h_{ww} & h_{wx} & h_{wy} \
h_{xv} & h_{xw} & h_{xv} & h_{xy} \
h_{yv} & h_{yw} & h_{yx} & h_{yy}
end{pmatrix}.
$$
By the equality of mixed partials the eigenvalues are real and the equilibrium point will be a minimum if each eigenvalue is positive, a maximum if they are all negative, and a saddle if they have different signs. $0$ eigenvalue is a still an indeterminate case.
You will also have to be careful with your domain. Since it does not contain the hyperplane $w=0$, it is possible that the extrema will lie on this hyperplane but your function cannot reach it in this domain.
answered Nov 30 '18 at 2:17
whpowell96whpowell96
53115
add a comment |
The equilibrium points will be where the gradient $nabla h = (h_v,h_w,h_x,h_y)$ is the $0$ vector. To tell whether or not it is a maximum/minimum/saddle, you have to look at the eigenvalues of the Hessian matrix,
$$
H = begin{pmatrix}
h_{vv} & h_{vw} & h_{vx} & h_{vy} \
h_{wv} & h_{ww} & h_{wx} & h_{wy} \
h_{xv} & h_{xw} & h_{xv} & h_{xy} \
h_{yv} & h_{yw} & h_{yx} & h_{yy}
end{pmatrix}.
$$
By the equality of mixed partials the eigenvalues are real and the equilibrium point will be a minimum if each eigenvalue is positive, a maximum if they are all negative, and a saddle if they have different signs. $0$ eigenvalue is a still an indeterminate case.
You will also have to be careful with your domain. Since it does not contain the hyperplane $w=0$, it is possible that the extrema will lie on this hyperplane but your function cannot reach it in this domain.
answered Nov 30 '18 at 2:17
whpowell96whpowell96
53115
The equilibrium points will be where the gradient $nabla h = (h_v,h_w,h_x,h_y)$ is the $0$ vector. To tell whether or not it is a maximum/minimum/saddle, you have to look at the eigenvalues of the Hessian matrix,
$$
H = begin{pmatrix}
h_{vv} & h_{vw} & h_{vx} & h_{vy} \
h_{wv} & h_{ww} & h_{wx} & h_{wy} \
h_{xv} & h_{xw} & h_{xv} & h_{xy} \
h_{yv} & h_{yw} & h_{yx} & h_{yy}
end{pmatrix}.
$$
By the equality of mixed partials the eigenvalues are real and the equilibrium point will be a minimum if each eigenvalue is positive, a maximum if they are all negative, and a saddle if they have different signs. $0$ eigenvalue is a still an indeterminate case.
You will also have to be careful with your domain. Since it does not contain the hyperplane $w=0$, it is possible that the extrema will lie on this hyperplane but your function cannot reach it in this domain.
answered Nov 30 '18 at 2:17
whpowell96whpowell96
53115
answered Nov 30 '18 at 2:17
whpowell96whpowell96
53115
answered Nov 30 '18 at 2:17
whpowell96whpowell96
53115
answered Nov 30 '18 at 2:17
whpowell96whpowell96
53115
53115
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%2f3019514%2fmaxima-minima-hv-w-x-y-6v2-12varctanw-frac12w-expx2x2y2%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
