Inequality with extreme values











up vote
0
down vote

favorite












$f:[0,1]rightarrow mathbb{R}$ is continuous and non-constant function. $F$ is the indefinite integral of $f$ such that $F(0)=F(1)=0$. $m$ and $M$ are the minimum and the maximum values of $f$. Now we define $g:[0,1]rightarrow mathbb{R},: g(x)=xf(x)$ and the indefinite integral of $g$ is $G$, such that $G(0)=0$. Prove that $|G(1)|leqdfrac{−mM}{2(M−m)}$ . Can somebody help me,please?






calculus functional-analysis functions indefinite-integrals extreme-value-analysis







share|cite|improve this question




















  • 1




    Is $g(x)$ supposed to be $xf(x)$ or $xF(x)$?
    – Kavi Rama Murthy
    2 hours ago










  • g(x) is xf(x)..
    – Andrei Gabor
    55 mins ago










  • In that case the question has nothing to do with $F$. I suggest that reference to $F$ be removed from the question.
    – Kavi Rama Murthy
    52 mins ago















up vote
0
down vote

favorite












$f:[0,1]rightarrow mathbb{R}$ is continuous and non-constant function. $F$ is the indefinite integral of $f$ such that $F(0)=F(1)=0$. $m$ and $M$ are the minimum and the maximum values of $f$. Now we define $g:[0,1]rightarrow mathbb{R},: g(x)=xf(x)$ and the indefinite integral of $g$ is $G$, such that $G(0)=0$. Prove that $|G(1)|leqdfrac{−mM}{2(M−m)}$ . Can somebody help me,please?






calculus functional-analysis functions indefinite-integrals extreme-value-analysis







share|cite|improve this question




















  • 1




    Is $g(x)$ supposed to be $xf(x)$ or $xF(x)$?
    – Kavi Rama Murthy
    2 hours ago










  • g(x) is xf(x)..
    – Andrei Gabor
    55 mins ago










  • In that case the question has nothing to do with $F$. I suggest that reference to $F$ be removed from the question.
    – Kavi Rama Murthy
    52 mins ago













up vote
0
down vote

favorite









up vote
0
down vote

favorite











$f:[0,1]rightarrow mathbb{R}$ is continuous and non-constant function. $F$ is the indefinite integral of $f$ such that $F(0)=F(1)=0$. $m$ and $M$ are the minimum and the maximum values of $f$. Now we define $g:[0,1]rightarrow mathbb{R},: g(x)=xf(x)$ and the indefinite integral of $g$ is $G$, such that $G(0)=0$. Prove that $|G(1)|leqdfrac{−mM}{2(M−m)}$ . Can somebody help me,please?






calculus functional-analysis functions indefinite-integrals extreme-value-analysis







share|cite|improve this question















$f:[0,1]rightarrow mathbb{R}$ is continuous and non-constant function. $F$ is the indefinite integral of $f$ such that $F(0)=F(1)=0$. $m$ and $M$ are the minimum and the maximum values of $f$. Now we define $g:[0,1]rightarrow mathbb{R},: g(x)=xf(x)$ and the indefinite integral of $g$ is $G$, such that $G(0)=0$. Prove that $|G(1)|leqdfrac{−mM}{2(M−m)}$ . Can somebody help me,please?






calculus functional-analysis functions indefinite-integrals extreme-value-analysis




calculus functional-analysis functions indefinite-integrals extreme-value-analysis






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 2 hours ago









Yadati Kiran

348110




348110










asked 2 hours ago









Andrei Gabor

576




576








  • 1




    Is $g(x)$ supposed to be $xf(x)$ or $xF(x)$?
    – Kavi Rama Murthy
    2 hours ago










  • g(x) is xf(x)..
    – Andrei Gabor
    55 mins ago










  • In that case the question has nothing to do with $F$. I suggest that reference to $F$ be removed from the question.
    – Kavi Rama Murthy
    52 mins ago














  • 1




    Is $g(x)$ supposed to be $xf(x)$ or $xF(x)$?
    – Kavi Rama Murthy
    2 hours ago










  • g(x) is xf(x)..
    – Andrei Gabor
    55 mins ago










  • In that case the question has nothing to do with $F$. I suggest that reference to $F$ be removed from the question.
    – Kavi Rama Murthy
    52 mins ago








1




1




Is $g(x)$ supposed to be $xf(x)$ or $xF(x)$?
– Kavi Rama Murthy
2 hours ago




Is $g(x)$ supposed to be $xf(x)$ or $xF(x)$?
– Kavi Rama Murthy
2 hours ago












g(x) is xf(x)..
– Andrei Gabor
55 mins ago




g(x) is xf(x)..
– Andrei Gabor
55 mins ago












In that case the question has nothing to do with $F$. I suggest that reference to $F$ be removed from the question.
– Kavi Rama Murthy
52 mins ago




In that case the question has nothing to do with $F$. I suggest that reference to $F$ be removed from the question.
– Kavi Rama Murthy
52 mins ago















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%2f3004659%2finequality-with-extreme-values%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%2f3004659%2finequality-with-extreme-values%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

MongoDB - Not Authorized To Execute Command

Image
.everyoneloves__top-leaderboard:empty,.everyoneloves__mid-leaderboard:empty,.everyoneloves__bot-mid-leaderboard:empty{ height:90px;width:728px;box-sizing:border-box; } 0 I have successfully enabled authorization on MongoDB and I have created an account on the admin database and then I created an account for my database called test. The following connection string to connect to my test database works successfully: mongo --host 192.168.17.52 --port 27017 -u user1 -p password --authenticationDatabase test Only problem I have now is, I cannot execute commands such as: show dbs. I get the following error when I try to do so: "errmsg" : "not authorized on admin to execute command { listDatabases: 1.0, lsid: { id: UUID("a1d5bc0d-bc58-485e-b232-270758a89455") }, $db: "admin...
Read more

How to fix TextFormField cause rebuild widget in Flutter

Image
.everyoneloves__top-leaderboard:empty,.everyoneloves__mid-leaderboard:empty,.everyoneloves__bot-mid-leaderboard:empty{ height:90px;width:728px;box-sizing:border-box; } 0 I've got the same issue at #9280. I tried all solutions from that topic with/without GlobalKey for Scaffold and Form but the problem wasn't fixed. I tried to build a authentication form with login and signup. But the keyboard doesn't show when I tap the TextFormField at the first time. Then I try to input some letter from physical keyboard, the widget is rebuilt. Like this: login screen Nothing to be log when I run flutter run --verbose, so I logged manually. Every tapping on field, the widget called dispose -> init -> build. [√] Flutter (Channel beta, v1.0.0, on Microsoft Windows [Version 10.0.17763.195], ...
Read more

in spring boot 2.1 many test slices are not allowed anymore due to multiple @BootstrapWith

Image
1 I tried to upgrade a yummy sandwich made of two test slices (@JsonTest and @JdbcTest in my case, crunchy test code in between) adding spring boot 2.1 flavour to it. But it seems it was not much of a success. I cannot annotate my tests with many @...Test since they are now each bringing their own XxxTestContextBootstrapper. It used to work when they all used same SpringBootTestContextBootstrapper. @RunWith(SpringRunner.class) @JdbcTest @JsonTest public class Test { @Test public void test() { System.out.printn("Hello, World !"); } } The error I get from BootstrapUtils is illegalStateException : Configuration error: found multiple declarations of @BootstrapWith for test class I understand I might be doing something wrong here but is there an easy way I could load both Json and Jdbc contex...
Read more