Arithmetic mean geometric mean inequality unclearproving inequality?Practicing the arithmetic-geometric means...

How can a function with a hole (removable discontinuity) equal a function with no hole?

Is expanding the research of a group into machine learning as a PhD student risky?

Trouble understanding the speech of overseas colleagues

Implement the Thanos sorting algorithm

Term for the "extreme-extension" version of a straw man fallacy?

Is there a korbon needed for conversion?

Where does the Z80 processor start executing from?

Did Dumbledore lie to Harry about how long he had James Potter's invisibility cloak when he was examining it? If so, why?

What is the difference between "behavior" and "behaviour"?

Lay out the Carpet

Do sorcerers' Subtle Spells require a skill check to be unseen?

What does "I’d sit this one out, Cap," imply or mean in the context?

A Rare Riley Riddle

How does the UK government determine the size of a mandate?

How to pronounce the slash sign

How long to clear the 'suck zone' of a turbofan after start is initiated?

Failed to fetch jessie backports repository

Why not increase contact surface when reentering the atmosphere?

How can I kill an app using Terminal?

India just shot down a satellite from the ground. At what altitude range is the resulting debris field?

How do I rename a Linux host without needing to reboot for the rename to take effect?

Escape a backup date in a file name

Is there a good way to store credentials outside of a password manager?

Can the discrete variable be a negative number?



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 $frac{a}{b+c}+frac{b}{c+a}+frac{c}{a+b}geqfrac{3}{2}$Problem in Arithmetic Mean - Geometric Mean inequalityProving Cauchy-Schwarz with Arithmetic Geometric meanInequality involving a kind of Harmonic mean













1












$begingroup$


I know that the AM-GM inequality takes the form $$ frac{x + y}{2} geq sqrt{xy},$$ but I read in a book another form which is $$ frac{x^2 + y^2}{2} geq |xy|,$$ but I am wondering how the second comes from the first? could anyone explain this for me please?






calculus inequality







share|cite|improve this question











$endgroup$

















    1












    $begingroup$


    I know that the AM-GM inequality takes the form $$ frac{x + y}{2} geq sqrt{xy},$$ but I read in a book another form which is $$ frac{x^2 + y^2}{2} geq |xy|,$$ but I am wondering how the second comes from the first? could anyone explain this for me please?






    calculus inequality







    share|cite|improve this question











    $endgroup$















      1












      1








      1


      1



      $begingroup$


      I know that the AM-GM inequality takes the form $$ frac{x + y}{2} geq sqrt{xy},$$ but I read in a book another form which is $$ frac{x^2 + y^2}{2} geq |xy|,$$ but I am wondering how the second comes from the first? could anyone explain this for me please?






      calculus inequality







      share|cite|improve this question











      $endgroup$




      I know that the AM-GM inequality takes the form $$ frac{x + y}{2} geq sqrt{xy},$$ but I read in a book another form which is $$ frac{x^2 + y^2}{2} geq |xy|,$$ but I am wondering how the second comes from the first? could anyone explain this for me please?






      calculus inequality




      calculus inequality






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited 5 hours ago









      Bernard

      123k741117




      123k741117










      asked 5 hours ago









      hopefullyhopefully

      274114




      274114






















          2 Answers
          2






          active

          oldest

          votes


















          4












          $begingroup$

          If you plug $x=X^2$, $y=Y^2$ into the first inequality you get
          $$frac{X^2+Y^2}{2} ge sqrt{X^2Y^2} = sqrt{(XY)^2}=|XY|,$$
          which is the second inequality (modulo capitalization).






          share|cite|improve this answer









          $endgroup$





















            3












            $begingroup$

            The AM-GM inequality for $n$ non-negative values is



            $frac1{n}(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



            $frac1{n}(sum_{k=1}^n y_k^n)
            ge prod_{k=1}^n y_k
            $            .



            It is useful to recognize
            these disguises.






            share|cite|improve this answer









            $endgroup$













              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


















              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









              4












              $begingroup$

              If you plug $x=X^2$, $y=Y^2$ into the first inequality you get
              $$frac{X^2+Y^2}{2} ge sqrt{X^2Y^2} = sqrt{(XY)^2}=|XY|,$$
              which is the second inequality (modulo capitalization).






              share|cite|improve this answer









              $endgroup$


















                4












                $begingroup$

                If you plug $x=X^2$, $y=Y^2$ into the first inequality you get
                $$frac{X^2+Y^2}{2} ge sqrt{X^2Y^2} = sqrt{(XY)^2}=|XY|,$$
                which is the second inequality (modulo capitalization).






                share|cite|improve this answer









                $endgroup$
















                  4












                  4








                  4





                  $begingroup$

                  If you plug $x=X^2$, $y=Y^2$ into the first inequality you get
                  $$frac{X^2+Y^2}{2} ge sqrt{X^2Y^2} = sqrt{(XY)^2}=|XY|,$$
                  which is the second inequality (modulo capitalization).






                  share|cite|improve this answer









                  $endgroup$



                  If you plug $x=X^2$, $y=Y^2$ into the first inequality you get
                  $$frac{X^2+Y^2}{2} ge sqrt{X^2Y^2} = sqrt{(XY)^2}=|XY|,$$
                  which is the second inequality (modulo capitalization).







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered 5 hours ago









                  jgonjgon

                  16k32143




                  16k32143























                      3












                      $begingroup$

                      The AM-GM inequality for $n$ non-negative values is



                      $frac1{n}(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



                      $frac1{n}(sum_{k=1}^n y_k^n)
                      ge prod_{k=1}^n y_k
                      $                      .



                      It is useful to recognize
                      these disguises.






                      share|cite|improve this answer









                      $endgroup$


















                        3












                        $begingroup$

                        The AM-GM inequality for $n$ non-negative values is



                        $frac1{n}(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



                        $frac1{n}(sum_{k=1}^n y_k^n)
                        ge prod_{k=1}^n y_k
                        $                        .



                        It is useful to recognize
                        these disguises.






                        share|cite|improve this answer









                        $endgroup$
















                          3












                          3








                          3





                          $begingroup$

                          The AM-GM inequality for $n$ non-negative values is



                          $frac1{n}(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



                          $frac1{n}(sum_{k=1}^n y_k^n)
                          ge prod_{k=1}^n y_k
                          $                          .



                          It is useful to recognize
                          these disguises.






                          share|cite|improve this answer









                          $endgroup$



                          The AM-GM inequality for $n$ non-negative values is



                          $frac1{n}(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



                          $frac1{n}(sum_{k=1}^n y_k^n)
                          ge prod_{k=1}^n y_k
                          $                          .



                          It is useful to recognize
                          these disguises.







                          share|cite|improve this answer












                          share|cite|improve this answer



                          share|cite|improve this answer










                          answered 5 hours ago









                          marty cohenmarty cohen

                          74.9k549130




                          74.9k549130






























                              draft saved

                              draft discarded




















































                              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.




                              draft saved


                              draft discarded














                              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





















































                              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

                              Couldn't open a raw socket. Error: Permission denied (13) (nmap)Is it possible to run networking commands...

                              Image
                              Why didn't Boeing produce its own regional jet? What is the most common color to indicate the input-field is disabled? Theorists sure want true answers to this! What is the opposite of "eschatology"? What do you call someone who asks many questions? Is there a hemisphere-neutral way of specifying a season? Is this draw by repetition? Bug? String.split chopping off empty trailing value Is it inappropriate for a student to attend their mentor's dissertation defense? How dangerous is XSS Why do I get negative height? Is it possible to map
                              Read more

                              VNC viewer RFB protocol error: bad desktop size 0x0I Cannot Type the Key 'd' (lowercase) in VNC Viewer...

                              Image
                              risk of flooding in petra in november How to get directions in deep space? How could a planet have erratic days? 15% tax on $7.5k earnings. Is that right? Did the UK lift the requirement for registering SIM cards? Is it allowed to activate the ability of multiple planeswalkers in a single turn? Stack Interview Code methods made from class Node and Smart Pointers Non-trope happy ending? What is the highest possible scrabble score for placing a single tile Open a doc from terminal, but not by its name The IT department bottlenecks progress, how should I handle this?
                              Read more

                              Why not use the yoke to control yaw, as well as pitch and roll? Announcing the arrival of...

                              Image
                              Does the Mueller report show a conspiracy between Russia and the Trump Campaign? Relating to the President and obstruction, were Mueller's conclusions preordained? How much damage would a cupful of neutron star matter do to the Earth? Nose gear failure in single prop aircraft: belly landing or nose landing? Should a wizard buy fine inks every time he want to copy spells into his spellbook? Can you force honesty by using the Speak with Dead and Zone of Truth spells together? A trigonometry question from STEP examination Special flights What is the "studentd" process? systemd and copy (/bin/cp): no such file or directory
                              Read more