Arithmetic mean geometric mean inequality unclearproving inequality?Practicing the arithmetic-geometric means inequalityArithmetic Mean and Geometric Mean Question, Guidance NeededHow prove Reversing the Arithmetic mean – Geometric mean inequality？Mean Value Theorem and Inequality.Using arithmetic mean>geometric meanNesbitt's Inequality $fracab+c+fracbc+a+fracca+bgeqfrac32$Problem in Arithmetic Mean - Geometric Mean inequalityProving Cauchy-Schwarz with Arithmetic Geometric meanInequality involving a kind of Harmonic mean
Arithmetic mean geometric mean inequality unclear
I'm in charge of equipment buying but no one's ever happy with what I choose. How to fix this?
Is there a good way to store credentials outside of a password manager?
Return the Closest Prime Number
Tiptoe or tiphoof? Adjusting words to better fit fantasy races
Why, precisely, is argon used in neutrino experiments?
How can I kill an app using Terminal?
Would this custom Sorcerer variant that can only learn any verbal-component-only spell be unbalanced?
How to safely derail a train during transit?
Two monoidal structures and copowering
Why not increase contact surface when reentering the atmosphere?
What is the intuitive meaning of having a linear relationship between the logs of two variables?
Was Spock the First Vulcan in Starfleet?
How do I go from 300 unfinished/half written blog posts, to published posts?
Would a high gravity rocky planet be guaranteed to have an atmosphere?
Is a stroke of luck acceptable after a series of unfavorable events?
How long to clear the 'suck zone' of a turbofan after start is initiated?
Short story about space worker geeks who zone out by 'listening' to radiation from stars
Class Action - which options I have?
What does "I’d sit this one out, Cap," imply or mean in the context?
Applicability of Single Responsibility Principle
Why escape if the_content isnt?
How does it work when somebody invests in my business?
Is exact Kanji stroke length important?
Arithmetic mean geometric mean inequality unclear
proving inequality?Practicing the arithmetic-geometric means inequalityArithmetic Mean and Geometric Mean Question, Guidance NeededHow prove Reversing the Arithmetic mean – Geometric mean inequality？Mean Value Theorem and Inequality.Using arithmetic mean>geometric meanNesbitt's Inequality $fracab+c+fracbc+a+fracca+bgeqfrac32$Problem in Arithmetic Mean - Geometric Mean inequalityProving Cauchy-Schwarz with Arithmetic Geometric meanInequality involving a kind of Harmonic mean
$begingroup$
I know that the AM-GM inequality takes the form $$ fracx + y2 geq sqrtxy,$$ but I read in a book another form which is $$ fracx^2 + y^22 geq |xy|,$$ but I am wondering how the second comes from the first? could anyone explain this for me please?
calculus inequality
edited 5 hours ago
Bernard
123k741117
asked 5 hours ago
hopefullyhopefully
274114
$endgroup$
add a comment |
$begingroup$
I know that the AM-GM inequality takes the form $$ fracx + y2 geq sqrtxy,$$ but I read in a book another form which is $$ fracx^2 + y^22 geq |xy|,$$ but I am wondering how the second comes from the first? could anyone explain this for me please?
calculus inequality
edited 5 hours ago
Bernard
123k741117
asked 5 hours ago
hopefullyhopefully
274114
$endgroup$
add a comment |
$begingroup$
I know that the AM-GM inequality takes the form $$ fracx + y2 geq sqrtxy,$$ but I read in a book another form which is $$ fracx^2 + y^22 geq |xy|,$$ but I am wondering how the second comes from the first? could anyone explain this for me please?
calculus inequality
edited 5 hours ago
Bernard
123k741117
asked 5 hours ago
hopefullyhopefully
274114
$endgroup$
I know that the AM-GM inequality takes the form $$ fracx + y2 geq sqrtxy,$$ but I read in a book another form which is $$ fracx^2 + y^22 geq |xy|,$$ but I am wondering how the second comes from the first? could anyone explain this for me please?
calculus inequality
calculus inequality
edited 5 hours ago
Bernard
123k741117
asked 5 hours ago
hopefullyhopefully
274114
edited 5 hours ago
Bernard
123k741117
asked 5 hours ago
hopefullyhopefully
274114
edited 5 hours ago
Bernard
123k741117
edited 5 hours ago
Bernard
123k741117
edited 5 hours ago
Bernard
123k741117
123k741117
asked 5 hours ago
hopefullyhopefully
274114
asked 5 hours ago
hopefullyhopefully
274114
asked 5 hours ago
hopefullyhopefully
274114
274114
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
If you plug $x=X^2$, $y=Y^2$ into the first inequality you get
$$fracX^2+Y^22 ge sqrtX^2Y^2 = sqrt(XY)^2=|XY|,$$
which is the second inequality (modulo capitalization).
answered 5 hours ago
jgonjgon
16k32143
$endgroup$
add a comment |
$begingroup$
The AM-GM inequality for $n$ non-negative values is
$frac1n(sum_k=1^n x_k)
ge (prod_k=1^n x_k)^1/n
$.
This can be rewritten in two ways.
First,
by simple algebra,
$(sum_k=1^n x_i)^n
ge n^n(prod_k=1^n x_k)
$.
Second,
letting $x_k = y_k^n$,
this becomes
$frac1n(sum_k=1^n y_k^n)
ge prod_k=1^n y_k
$.
It is useful to recognize
these disguises.
answered 5 hours ago
marty cohenmarty cohen
74.9k549130
$endgroup$
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
);
);
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%2f3165273%2farithmetic-mean-geometric-mean-inequality-unclear%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
If you plug $x=X^2$, $y=Y^2$ into the first inequality you get
$$fracX^2+Y^22 ge sqrtX^2Y^2 = sqrt(XY)^2=|XY|,$$
which is the second inequality (modulo capitalization).
answered 5 hours ago
jgonjgon
16k32143
$endgroup$
add a comment |
$begingroup$
If you plug $x=X^2$, $y=Y^2$ into the first inequality you get
$$fracX^2+Y^22 ge sqrtX^2Y^2 = sqrt(XY)^2=|XY|,$$
which is the second inequality (modulo capitalization).
answered 5 hours ago
jgonjgon
16k32143
$endgroup$
add a comment |
$begingroup$
If you plug $x=X^2$, $y=Y^2$ into the first inequality you get
$$fracX^2+Y^22 ge sqrtX^2Y^2 = sqrt(XY)^2=|XY|,$$
which is the second inequality (modulo capitalization).
answered 5 hours ago
jgonjgon
16k32143
$endgroup$
If you plug $x=X^2$, $y=Y^2$ into the first inequality you get
$$fracX^2+Y^22 ge sqrtX^2Y^2 = sqrt(XY)^2=|XY|,$$
which is the second inequality (modulo capitalization).
answered 5 hours ago
jgonjgon
16k32143
answered 5 hours ago
jgonjgon
16k32143
answered 5 hours ago
jgonjgon
16k32143
answered 5 hours ago
jgonjgon
16k32143
16k32143
add a comment |
add a comment |
$begingroup$
The AM-GM inequality for $n$ non-negative values is
$frac1n(sum_k=1^n x_k)
ge (prod_k=1^n x_k)^1/n
$.
This can be rewritten in two ways.
First,
by simple algebra,
$(sum_k=1^n x_i)^n
ge n^n(prod_k=1^n x_k)
$.
Second,
letting $x_k = y_k^n$,
this becomes
$frac1n(sum_k=1^n y_k^n)
ge prod_k=1^n y_k
$.
It is useful to recognize
these disguises.
answered 5 hours ago
marty cohenmarty cohen
74.9k549130
$endgroup$
add a comment |
$begingroup$
The AM-GM inequality for $n$ non-negative values is
$frac1n(sum_k=1^n x_k)
ge (prod_k=1^n x_k)^1/n
$.
This can be rewritten in two ways.
First,
by simple algebra,
$(sum_k=1^n x_i)^n
ge n^n(prod_k=1^n x_k)
$.
Second,
letting $x_k = y_k^n$,
this becomes
$frac1n(sum_k=1^n y_k^n)
ge prod_k=1^n y_k
$.
It is useful to recognize
these disguises.
answered 5 hours ago
marty cohenmarty cohen
74.9k549130
$endgroup$
add a comment |
$begingroup$
The AM-GM inequality for $n$ non-negative values is
$frac1n(sum_k=1^n x_k)
ge (prod_k=1^n x_k)^1/n
$.
This can be rewritten in two ways.
First,
by simple algebra,
$(sum_k=1^n x_i)^n
ge n^n(prod_k=1^n x_k)
$.
Second,
letting $x_k = y_k^n$,
this becomes
$frac1n(sum_k=1^n y_k^n)
ge prod_k=1^n y_k
$.
It is useful to recognize
these disguises.
answered 5 hours ago
marty cohenmarty cohen
74.9k549130
$endgroup$
The AM-GM inequality for $n$ non-negative values is
$frac1n(sum_k=1^n x_k)
ge (prod_k=1^n x_k)^1/n
$.
This can be rewritten in two ways.
First,
by simple algebra,
$(sum_k=1^n x_i)^n
ge n^n(prod_k=1^n x_k)
$.
Second,
letting $x_k = y_k^n$,
this becomes
$frac1n(sum_k=1^n y_k^n)
ge prod_k=1^n y_k
$.
It is useful to recognize
these disguises.
answered 5 hours ago
marty cohenmarty cohen
74.9k549130
answered 5 hours ago
marty cohenmarty cohen
74.9k549130
answered 5 hours ago
marty cohenmarty cohen
74.9k549130
answered 5 hours ago
marty cohenmarty cohen
74.9k549130
74.9k549130
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.
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%2f3165273%2farithmetic-mean-geometric-mean-inequality-unclear%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
