calculus lll how to prove this question?











up vote
-2
down vote

favorite












it is known that $f(0,0) = 0$ and $f$ is differentiable at $(0,0)$. It is also known that for every $t>0$ we have $f(cos(t)/t , sin(t)/t) > 0$.



Show that necessarily $operatorname{Gradient}(0,0) = (0,0)$.



I have no clue how to solve this. I have tried to show through limits, I have tried to get something from the definition ( that with $varepsilon, :(x/y) to 0$ at the limit)



I am sorry for the bad latex language really.










share|cite|improve this question
























  • please help i have asked this question before (a week ago) but no one try to help thank you
    – Razi Awad
    Nov 18 at 14:41












  • nevermind i'll do it myself remove this post.
    – Razi Awad
    Nov 18 at 14:49






  • 1




    Is your function defined as $$f: mathbb{R} to mathbb{R}^2, t mapsto begin{cases} left(frac{cos(t)}{t}, frac{sin(t)}{t}right) & t > 0 \ 0 & t = 0 end{cases}$$?
    – Viktor Glombik
    Nov 18 at 14:49










  • yes exactly like that
    – Razi Awad
    Nov 18 at 14:50










  • do you know how the gradient is defined?
    – Viktor Glombik
    Nov 18 at 14:52















up vote
-2
down vote

favorite












it is known that $f(0,0) = 0$ and $f$ is differentiable at $(0,0)$. It is also known that for every $t>0$ we have $f(cos(t)/t , sin(t)/t) > 0$.



Show that necessarily $operatorname{Gradient}(0,0) = (0,0)$.



I have no clue how to solve this. I have tried to show through limits, I have tried to get something from the definition ( that with $varepsilon, :(x/y) to 0$ at the limit)



I am sorry for the bad latex language really.










share|cite|improve this question
























  • please help i have asked this question before (a week ago) but no one try to help thank you
    – Razi Awad
    Nov 18 at 14:41












  • nevermind i'll do it myself remove this post.
    – Razi Awad
    Nov 18 at 14:49






  • 1




    Is your function defined as $$f: mathbb{R} to mathbb{R}^2, t mapsto begin{cases} left(frac{cos(t)}{t}, frac{sin(t)}{t}right) & t > 0 \ 0 & t = 0 end{cases}$$?
    – Viktor Glombik
    Nov 18 at 14:49










  • yes exactly like that
    – Razi Awad
    Nov 18 at 14:50










  • do you know how the gradient is defined?
    – Viktor Glombik
    Nov 18 at 14:52













up vote
-2
down vote

favorite









up vote
-2
down vote

favorite











it is known that $f(0,0) = 0$ and $f$ is differentiable at $(0,0)$. It is also known that for every $t>0$ we have $f(cos(t)/t , sin(t)/t) > 0$.



Show that necessarily $operatorname{Gradient}(0,0) = (0,0)$.



I have no clue how to solve this. I have tried to show through limits, I have tried to get something from the definition ( that with $varepsilon, :(x/y) to 0$ at the limit)



I am sorry for the bad latex language really.










share|cite|improve this question















it is known that $f(0,0) = 0$ and $f$ is differentiable at $(0,0)$. It is also known that for every $t>0$ we have $f(cos(t)/t , sin(t)/t) > 0$.



Show that necessarily $operatorname{Gradient}(0,0) = (0,0)$.



I have no clue how to solve this. I have tried to show through limits, I have tried to get something from the definition ( that with $varepsilon, :(x/y) to 0$ at the limit)



I am sorry for the bad latex language really.







multivariable-calculus






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Nov 18 at 15:02









David K

51.4k340113




51.4k340113










asked Nov 18 at 14:40









Razi Awad

186




186












  • please help i have asked this question before (a week ago) but no one try to help thank you
    – Razi Awad
    Nov 18 at 14:41












  • nevermind i'll do it myself remove this post.
    – Razi Awad
    Nov 18 at 14:49






  • 1




    Is your function defined as $$f: mathbb{R} to mathbb{R}^2, t mapsto begin{cases} left(frac{cos(t)}{t}, frac{sin(t)}{t}right) & t > 0 \ 0 & t = 0 end{cases}$$?
    – Viktor Glombik
    Nov 18 at 14:49










  • yes exactly like that
    – Razi Awad
    Nov 18 at 14:50










  • do you know how the gradient is defined?
    – Viktor Glombik
    Nov 18 at 14:52


















  • please help i have asked this question before (a week ago) but no one try to help thank you
    – Razi Awad
    Nov 18 at 14:41












  • nevermind i'll do it myself remove this post.
    – Razi Awad
    Nov 18 at 14:49






  • 1




    Is your function defined as $$f: mathbb{R} to mathbb{R}^2, t mapsto begin{cases} left(frac{cos(t)}{t}, frac{sin(t)}{t}right) & t > 0 \ 0 & t = 0 end{cases}$$?
    – Viktor Glombik
    Nov 18 at 14:49










  • yes exactly like that
    – Razi Awad
    Nov 18 at 14:50










  • do you know how the gradient is defined?
    – Viktor Glombik
    Nov 18 at 14:52
















please help i have asked this question before (a week ago) but no one try to help thank you
– Razi Awad
Nov 18 at 14:41






please help i have asked this question before (a week ago) but no one try to help thank you
– Razi Awad
Nov 18 at 14:41














nevermind i'll do it myself remove this post.
– Razi Awad
Nov 18 at 14:49




nevermind i'll do it myself remove this post.
– Razi Awad
Nov 18 at 14:49




1




1




Is your function defined as $$f: mathbb{R} to mathbb{R}^2, t mapsto begin{cases} left(frac{cos(t)}{t}, frac{sin(t)}{t}right) & t > 0 \ 0 & t = 0 end{cases}$$?
– Viktor Glombik
Nov 18 at 14:49




Is your function defined as $$f: mathbb{R} to mathbb{R}^2, t mapsto begin{cases} left(frac{cos(t)}{t}, frac{sin(t)}{t}right) & t > 0 \ 0 & t = 0 end{cases}$$?
– Viktor Glombik
Nov 18 at 14:49












yes exactly like that
– Razi Awad
Nov 18 at 14:50




yes exactly like that
– Razi Awad
Nov 18 at 14:50












do you know how the gradient is defined?
– Viktor Glombik
Nov 18 at 14:52




do you know how the gradient is defined?
– Viktor Glombik
Nov 18 at 14:52















active

oldest

votes











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',
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


















StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3003612%2fcalculus-lll-how-to-prove-this-question%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown






























active

oldest

votes













active

oldest

votes









active

oldest

votes






active

oldest

votes
















 

draft saved


draft discarded



















































 


draft saved


draft discarded














StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3003612%2fcalculus-lll-how-to-prove-this-question%23new-answer', 'question_page');
}
);

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







Popular posts from this blog

Ellipse (mathématiques)

Quarter-circle Tiles

Mont Emei