Find the critical points of $g(x,y) = 4x^3 - 12xy + 3y^2 - 18y -5.$ The Next CEO of Stack OverflowFind all critical points of $f(x,y) = x^3 - 12xy + 8y^3$ and state maximum, minimum, or saddle points.Conceptual question: Critical PointsFind and classify the critical pointsFinding Critical Points and Local Maxima/Minima or Saddle PointHow would you find and classify ALL of the critical points of a function of 2 variables?Critical points have to be stationary?Find the critical points of the function $f(x,y)=(x^2+y^2)e^y^2-x^2$Determine and classify all critical points of function.Why are there critical points at these points?What is the nature of the critical points of $f(x,y) = 4x^2 -12xy + 9y^2$.
Example of a Mathematician/Physicist whose Other Publications during their PhD eclipsed their PhD Thesis
Why do airplanes bank sharply to the right after air-to-air refueling?
How many extra stops do monopods offer for tele photographs?
Can I use the load factor to estimate the lift?
Why don't programming languages automatically manage the synchronous/asynchronous problem?
Does increasing your ability score affect your main stat?
Is the D&D universe the same as the Forgotten Realms universe?
Why is the US ranked as #45 in Press Freedom ratings, despite its extremely permissive free speech laws?
Is wanting to ask what to write an indication that you need to change your story?
Axiom Schema vs Axiom
How to count occurrences of text in a file?
Would a grinding machine be a simple and workable propulsion system for an interplanetary spacecraft?
I want to delete every two lines after 3rd lines in file contain very large number of lines :
Why is my new battery behaving weirdly?
Would a completely good Muggle be able to use a wand?
"misplaced omit" error when >centering columns
WOW air has ceased operation, can I get my tickets refunded?
I believe this to be a fraud - hired, then asked to cash check and send cash as Bitcoin
How did people program for Consoles with multiple CPUs?
Is it okay to majorly distort historical facts while writing a fiction story?
Prepend last line of stdin to entire stdin
If Nick Fury and Coulson already knew about aliens (Kree and Skrull) why did they wait until Thor's appearance to start making weapons?
Necessary condition on homology group for a set to be contractible
Bartok - Syncopation (1): Meaning of notes in between Grand Staff
Find the critical points of $g(x,y) = 4x^3 - 12xy + 3y^2 - 18y -5.$
The Next CEO of Stack OverflowFind all critical points of $f(x,y) = x^3 - 12xy + 8y^3$ and state maximum, minimum, or saddle points.Conceptual question: Critical PointsFind and classify the critical pointsFinding Critical Points and Local Maxima/Minima or Saddle PointHow would you find and classify ALL of the critical points of a function of 2 variables?Critical points have to be stationary?Find the critical points of the function $f(x,y)=(x^2+y^2)e^y^2-x^2$Determine and classify all critical points of function.Why are there critical points at these points?What is the nature of the critical points of $f(x,y) = 4x^2 -12xy + 9y^2$.
$begingroup$
I have the function $g(x,y) = 4x^3 - 12xy + 3y^2 - 18y -5.$ The only critical points that I have found for this function are $(-1, 1)$, and $(3, 9)$. But my professor insisted that there are more critical points besides these two.
Can anyone help me find them please?
multivariable-calculus partial-derivative
edited Mar 27 at 19:37
thesmallprint
2,6711618
asked Mar 27 at 19:34
UchuukoUchuuko
156
$endgroup$
add a comment |
$begingroup$
I have the function $g(x,y) = 4x^3 - 12xy + 3y^2 - 18y -5.$ The only critical points that I have found for this function are $(-1, 1)$, and $(3, 9)$. But my professor insisted that there are more critical points besides these two.
Can anyone help me find them please?
multivariable-calculus partial-derivative
edited Mar 27 at 19:37
thesmallprint
2,6711618
asked Mar 27 at 19:34
UchuukoUchuuko
156
$endgroup$
$begingroup$
wolfram alpha gives that those two points you list are indeed the only critical points for $g$.
$endgroup$
– thesmallprint
Mar 27 at 19:42
$begingroup$
Okay, thank you. I will tell my professor.
$endgroup$
– Uchuuko
Mar 27 at 19:44
$begingroup$
Why not publish you answer so that it can be inspected closely?
$endgroup$
– NoChance
Mar 27 at 19:59
add a comment |
$begingroup$
I have the function $g(x,y) = 4x^3 - 12xy + 3y^2 - 18y -5.$ The only critical points that I have found for this function are $(-1, 1)$, and $(3, 9)$. But my professor insisted that there are more critical points besides these two.
Can anyone help me find them please?
multivariable-calculus partial-derivative
edited Mar 27 at 19:37
thesmallprint
2,6711618
asked Mar 27 at 19:34
UchuukoUchuuko
156
$endgroup$
I have the function $g(x,y) = 4x^3 - 12xy + 3y^2 - 18y -5.$ The only critical points that I have found for this function are $(-1, 1)$, and $(3, 9)$. But my professor insisted that there are more critical points besides these two.
Can anyone help me find them please?
multivariable-calculus partial-derivative
multivariable-calculus partial-derivative
edited Mar 27 at 19:37
thesmallprint
2,6711618
asked Mar 27 at 19:34
UchuukoUchuuko
156
edited Mar 27 at 19:37
thesmallprint
2,6711618
asked Mar 27 at 19:34
UchuukoUchuuko
156
edited Mar 27 at 19:37
thesmallprint
2,6711618
edited Mar 27 at 19:37
thesmallprint
2,6711618
edited Mar 27 at 19:37
thesmallprint
2,6711618
2,6711618
asked Mar 27 at 19:34
UchuukoUchuuko
156
asked Mar 27 at 19:34
UchuukoUchuuko
156
asked Mar 27 at 19:34
UchuukoUchuuko
156
156
$begingroup$
wolfram alpha gives that those two points you list are indeed the only critical points for $g$.
$endgroup$
– thesmallprint
Mar 27 at 19:42
$begingroup$
Okay, thank you. I will tell my professor.
$endgroup$
– Uchuuko
Mar 27 at 19:44
$begingroup$
Why not publish you answer so that it can be inspected closely?
$endgroup$
– NoChance
Mar 27 at 19:59
add a comment |
$begingroup$
wolfram alpha gives that those two points you list are indeed the only critical points for $g$.
$endgroup$
– thesmallprint
Mar 27 at 19:42
$begingroup$
Okay, thank you. I will tell my professor.
$endgroup$
– Uchuuko
Mar 27 at 19:44
$begingroup$
Why not publish you answer so that it can be inspected closely?
$endgroup$
– NoChance
Mar 27 at 19:59
$begingroup$
wolfram alpha gives that those two points you list are indeed the only critical points for $g$.
$endgroup$
– thesmallprint
Mar 27 at 19:42
$begingroup$
wolfram alpha gives that those two points you list are indeed the only critical points for $g$.
$endgroup$
– thesmallprint
Mar 27 at 19:42
$begingroup$
Okay, thank you. I will tell my professor.
$endgroup$
– Uchuuko
Mar 27 at 19:44
$begingroup$
Okay, thank you. I will tell my professor.
$endgroup$
– Uchuuko
Mar 27 at 19:44
$begingroup$
Why not publish you answer so that it can be inspected closely?
$endgroup$
– NoChance
Mar 27 at 19:59
$begingroup$
Why not publish you answer so that it can be inspected closely?
$endgroup$
– NoChance
Mar 27 at 19:59
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
The critical points occur where the gradient of the scalar field is zero. In this case
$nabla g(x,y)=(12x^2-12y, 6y-12x-18)=overrightarrow0$
If you solve this system of equations you'll find that the only two points are
$(-1, 1)$ and $(3, 9)$
As a consequence of the Fundamental Theorem of Algebra, these are the only two solutions.
answered Mar 27 at 20:01
officialnoriaofficialnoria
112
New contributor
officialnoria is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
$begingroup$
The system of equations can be interpreted as the intersection of a parabola and line, which can have at most two intersection points.
$endgroup$
– amd
Mar 27 at 20:04
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%2f3165010%2ffind-the-critical-points-of-gx-y-4x3-12xy-3y2-18y-5%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
$begingroup$
The critical points occur where the gradient of the scalar field is zero. In this case
$nabla g(x,y)=(12x^2-12y, 6y-12x-18)=overrightarrow0$
If you solve this system of equations you'll find that the only two points are
$(-1, 1)$ and $(3, 9)$
As a consequence of the Fundamental Theorem of Algebra, these are the only two solutions.
answered Mar 27 at 20:01
officialnoriaofficialnoria
112
New contributor
officialnoria is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
$begingroup$
The system of equations can be interpreted as the intersection of a parabola and line, which can have at most two intersection points.
$endgroup$
– amd
Mar 27 at 20:04
add a comment |
$begingroup$
The critical points occur where the gradient of the scalar field is zero. In this case
$nabla g(x,y)=(12x^2-12y, 6y-12x-18)=overrightarrow0$
If you solve this system of equations you'll find that the only two points are
$(-1, 1)$ and $(3, 9)$
As a consequence of the Fundamental Theorem of Algebra, these are the only two solutions.
answered Mar 27 at 20:01
officialnoriaofficialnoria
112
New contributor
officialnoria is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
$begingroup$
The system of equations can be interpreted as the intersection of a parabola and line, which can have at most two intersection points.
$endgroup$
– amd
Mar 27 at 20:04
add a comment |
$begingroup$
The critical points occur where the gradient of the scalar field is zero. In this case
$nabla g(x,y)=(12x^2-12y, 6y-12x-18)=overrightarrow0$
If you solve this system of equations you'll find that the only two points are
$(-1, 1)$ and $(3, 9)$
As a consequence of the Fundamental Theorem of Algebra, these are the only two solutions.
answered Mar 27 at 20:01
officialnoriaofficialnoria
112
New contributor
officialnoria is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
The critical points occur where the gradient of the scalar field is zero. In this case
$nabla g(x,y)=(12x^2-12y, 6y-12x-18)=overrightarrow0$
If you solve this system of equations you'll find that the only two points are
$(-1, 1)$ and $(3, 9)$
As a consequence of the Fundamental Theorem of Algebra, these are the only two solutions.
answered Mar 27 at 20:01
officialnoriaofficialnoria
112
New contributor
officialnoria is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
answered Mar 27 at 20:01
officialnoriaofficialnoria
112
New contributor
officialnoria is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
answered Mar 27 at 20:01
officialnoriaofficialnoria
112
answered Mar 27 at 20:01
officialnoriaofficialnoria
112
112
New contributor
officialnoria is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
officialnoria is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
officialnoria is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$begingroup$
The system of equations can be interpreted as the intersection of a parabola and line, which can have at most two intersection points.
$endgroup$
– amd
Mar 27 at 20:04
add a comment |
$begingroup$
The system of equations can be interpreted as the intersection of a parabola and line, which can have at most two intersection points.
$endgroup$
– amd
Mar 27 at 20:04
$begingroup$
The system of equations can be interpreted as the intersection of a parabola and line, which can have at most two intersection points.
$endgroup$
– amd
Mar 27 at 20:04
$begingroup$
The system of equations can be interpreted as the intersection of a parabola and line, which can have at most two intersection points.
$endgroup$
– amd
Mar 27 at 20:04
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.
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%2f3165010%2ffind-the-critical-points-of-gx-y-4x3-12xy-3y2-18y-5%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
$begingroup$
wolfram alpha gives that those two points you list are indeed the only critical points for $g$.
$endgroup$
– thesmallprint
Mar 27 at 19:42
$begingroup$
Okay, thank you. I will tell my professor.
$endgroup$
– Uchuuko
Mar 27 at 19:44
$begingroup$
Why not publish you answer so that it can be inspected closely?
$endgroup$
– NoChance
Mar 27 at 19:59