Fdtxjr

Can I make popcorn with any corn?

Problem of parity - Can we draw a closed path made up of 20 line segments...

In Japanese, what’s the difference between “Tonari ni” (となりに) and “Tsugi” (つぎ)? When would you use one over the other?

How can I make my BBEG immortal short of making them a Lich or Vampire?

Have astronauts in space suits ever taken selfies? If so, how?

Mathematical cryptic clues

Minkowski space

Accidentally leaked the solution to an assignment, what to do now? (I'm the prof)

LaTeX closing $ signs makes cursor jump

Why doesn't Newton's third law mean a person bounces back to where they started when they hit the ground?

Why are 150k or 200k jobs considered good when there are 300k+ births a month?

How is it possible to have an ability score that is less than 3?

Test whether all array elements are factors of a number

What is the word for reserving something for yourself before others do?

What does "Puller Prush Person" mean?

What defenses are there against being summoned by the Gate spell?

Do VLANs within a subnet need to have their own subnet for router on a stick?

What typically incentivizes a professor to change jobs to a lower ranking university?

How to find program name(s) of an installed package?

Can a Warlock become Neutral Good?

Languages that we cannot (dis)prove to be Context-Free

Which models of the Boeing 737 are still in production?

Why Is Death Allowed In the Matrix?

Watching something be written to a file live with tail

Find all local maximum and minimum points of the function $f$.

Maximum and Minimum ValueFinding all local maximum and minimum for some unusual functions.Finding intervals using local min and max (in interval notation form)Minimum/Maximum Question (Calculus)Local max, min and point of inflection of $y= ax^3 + 3bx^2 + 3cx - d$Find he local maximum and minimum value and saddle points of the function?Find the absolute minimum and maximum values of $f(theta) = cos theta$Maximum/MinimumDetermine local max., local min., and saddle points of the following function: $4x + 4y + x^2y + xy^2$$f$ differentiable $5$ times around $x=a, f'(a)=f''(a)=f'''(a)=0, f^(4)(x) <0 Rightarrow x=a$ is either a local minimum or a local maximum point

0

$begingroup$

Any help with this problem I have would be so much appreciated, i've been going round in circles and have no idea what I'm really doing.

I have the problem:

• Find all local maximum and minimum points of the function $f = xy$.

I've done the first derivative to get $$f' = 1(fracdydx)$$

But I have no clue on how to find the local max and min from this.
Any help would be grateful.

Thank you.

calculus maxima-minima

share|cite|improve this question

asked Mar 22 at 0:20

The StatisticianThe Statistician

115112

$endgroup$

add a comment | 

0

$begingroup$

Any help with this problem I have would be so much appreciated, i've been going round in circles and have no idea what I'm really doing.

I have the problem:

• Find all local maximum and minimum points of the function $f = xy$.

I've done the first derivative to get $$f' = 1(fracdydx)$$

But I have no clue on how to find the local max and min from this.
Any help would be grateful.

Thank you.

calculus maxima-minima

share|cite|improve this question

asked Mar 22 at 0:20

The StatisticianThe Statistician

115112

$endgroup$

add a comment | 

$begingroup$

Any help with this problem I have would be so much appreciated, i've been going round in circles and have no idea what I'm really doing.

I have the problem:

• Find all local maximum and minimum points of the function $f = xy$.

I've done the first derivative to get $$f' = 1(fracdydx)$$

But I have no clue on how to find the local max and min from this.
Any help would be grateful.

Thank you.

calculus maxima-minima

share|cite|improve this question

asked Mar 22 at 0:20

The StatisticianThe Statistician

115112

$endgroup$

share|cite|improve this question

asked Mar 22 at 0:20

The StatisticianThe Statistician

115112

share|cite|improve this question

asked Mar 22 at 0:20

The StatisticianThe Statistician

115112

asked Mar 22 at 0:20

The StatisticianThe Statistician

115112

asked Mar 22 at 0:20

The StatisticianThe Statistician

115112

1 Answer
1

active

oldest

votes

2

$begingroup$

The local extrema of a function occur at stationary points, i.e. where the function's total derivative is $0$. In terms of partial derivatives, if the function has continuous partial derivatives (which this one does), this is equivalent to the partial derivatives both being $0$ at the point.

Partially differentiating, we have
beginalign*
f_x &= y \
f_y &= x.
endalign*
These are both $0$ only at $(x, y) = (0, 0)$.

However, this stationary point is neither a local minimum nor maximum. Consider, for $t in BbbR setminus 0$,
beginalign*
f(t, t) &= t^2 > 0 = f(0, 0) \
f(t, -t) &= -t^2 < 0 = f(0, 0).
endalign*
That is, there are points as close as you want to $(0, 0)$ that have greater and lesser function values than $f(0, 0) = 0$.

share|cite|improve this answer

answered Mar 22 at 1:27

Theo BenditTheo Bendit

20.3k12353

$endgroup$

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
);



);

draft saved

draft discarded

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

Name

Email

Required, but never shown

StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3157574%2ffind-all-local-maximum-and-minimum-points-of-the-function-f%23new-answer', 'question_page');

);

Post as a guest

Name

Email

Required, but never shown

2

$begingroup$

The local extrema of a function occur at stationary points, i.e. where the function's total derivative is $0$. In terms of partial derivatives, if the function has continuous partial derivatives (which this one does), this is equivalent to the partial derivatives both being $0$ at the point.

Partially differentiating, we have
beginalign*
f_x &= y \
f_y &= x.
endalign*
These are both $0$ only at $(x, y) = (0, 0)$.

However, this stationary point is neither a local minimum nor maximum. Consider, for $t in BbbR setminus 0$,
beginalign*
f(t, t) &= t^2 > 0 = f(0, 0) \
f(t, -t) &= -t^2 < 0 = f(0, 0).
endalign*
That is, there are points as close as you want to $(0, 0)$ that have greater and lesser function values than $f(0, 0) = 0$.

share|cite|improve this answer

answered Mar 22 at 1:27

Theo BenditTheo Bendit

20.3k12353

$endgroup$

add a comment | 

2

$begingroup$

The local extrema of a function occur at stationary points, i.e. where the function's total derivative is $0$. In terms of partial derivatives, if the function has continuous partial derivatives (which this one does), this is equivalent to the partial derivatives both being $0$ at the point.

Partially differentiating, we have
beginalign*
f_x &= y \
f_y &= x.
endalign*
These are both $0$ only at $(x, y) = (0, 0)$.

However, this stationary point is neither a local minimum nor maximum. Consider, for $t in BbbR setminus 0$,
beginalign*
f(t, t) &= t^2 > 0 = f(0, 0) \
f(t, -t) &= -t^2 < 0 = f(0, 0).
endalign*
That is, there are points as close as you want to $(0, 0)$ that have greater and lesser function values than $f(0, 0) = 0$.

share|cite|improve this answer

answered Mar 22 at 1:27

Theo BenditTheo Bendit

20.3k12353

$endgroup$

add a comment | 

$begingroup$

The local extrema of a function occur at stationary points, i.e. where the function's total derivative is $0$. In terms of partial derivatives, if the function has continuous partial derivatives (which this one does), this is equivalent to the partial derivatives both being $0$ at the point.

Partially differentiating, we have
beginalign*
f_x &= y \
f_y &= x.
endalign*
These are both $0$ only at $(x, y) = (0, 0)$.

However, this stationary point is neither a local minimum nor maximum. Consider, for $t in BbbR setminus 0$,
beginalign*
f(t, t) &= t^2 > 0 = f(0, 0) \
f(t, -t) &= -t^2 < 0 = f(0, 0).
endalign*
That is, there are points as close as you want to $(0, 0)$ that have greater and lesser function values than $f(0, 0) = 0$.

share|cite|improve this answer

answered Mar 22 at 1:27

Theo BenditTheo Bendit

20.3k12353

$endgroup$

share|cite|improve this answer

answered Mar 22 at 1:27

Theo BenditTheo Bendit

20.3k12353

answered Mar 22 at 1:27

Theo BenditTheo Bendit

20.3k12353

answered Mar 22 at 1:27

Theo BenditTheo Bendit

20.3k12353