How to find minimum value












0












$begingroup$


For $a=sqrt{x^2-3sqrt2x+9}$ and $b=sqrt{x^2-5sqrt2x+25}$ what is the value of $x$ when $a+b$ is minimum and how to find this? Thanks in advance.






geometry optimization radicals maxima-minima







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    What have you tried?
    $endgroup$
    – Parcly Taxel
    Jan 28 at 9:42










  • $begingroup$
    I tried to differentiate it with respect to x
    $endgroup$
    – Kshitij Singh
    Jan 28 at 9:43










  • $begingroup$
    I tried to differentiate it with respect to x
    $endgroup$
    – Kshitij Singh
    Jan 28 at 9:43






  • 1




    $begingroup$
    Please typeset your equations using Mathjax for better presentation
    $endgroup$
    – Shubham Johri
    Jan 28 at 9:43
















0












$begingroup$


For $a=sqrt{x^2-3sqrt2x+9}$ and $b=sqrt{x^2-5sqrt2x+25}$ what is the value of $x$ when $a+b$ is minimum and how to find this? Thanks in advance.






geometry optimization radicals maxima-minima







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    What have you tried?
    $endgroup$
    – Parcly Taxel
    Jan 28 at 9:42










  • $begingroup$
    I tried to differentiate it with respect to x
    $endgroup$
    – Kshitij Singh
    Jan 28 at 9:43










  • $begingroup$
    I tried to differentiate it with respect to x
    $endgroup$
    – Kshitij Singh
    Jan 28 at 9:43






  • 1




    $begingroup$
    Please typeset your equations using Mathjax for better presentation
    $endgroup$
    – Shubham Johri
    Jan 28 at 9:43














0












0








0





$begingroup$


For $a=sqrt{x^2-3sqrt2x+9}$ and $b=sqrt{x^2-5sqrt2x+25}$ what is the value of $x$ when $a+b$ is minimum and how to find this? Thanks in advance.






geometry optimization radicals maxima-minima







share|cite|improve this question











$endgroup$




For $a=sqrt{x^2-3sqrt2x+9}$ and $b=sqrt{x^2-5sqrt2x+25}$ what is the value of $x$ when $a+b$ is minimum and how to find this? Thanks in advance.






geometry optimization radicals maxima-minima




geometry optimization radicals maxima-minima






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Jan 28 at 9:58









Michael Rozenberg

109k1896201




109k1896201










asked Jan 28 at 9:40









Kshitij SinghKshitij Singh

1106




1106








  • 1




    $begingroup$
    What have you tried?
    $endgroup$
    – Parcly Taxel
    Jan 28 at 9:42










  • $begingroup$
    I tried to differentiate it with respect to x
    $endgroup$
    – Kshitij Singh
    Jan 28 at 9:43










  • $begingroup$
    I tried to differentiate it with respect to x
    $endgroup$
    – Kshitij Singh
    Jan 28 at 9:43






  • 1




    $begingroup$
    Please typeset your equations using Mathjax for better presentation
    $endgroup$
    – Shubham Johri
    Jan 28 at 9:43














  • 1




    $begingroup$
    What have you tried?
    $endgroup$
    – Parcly Taxel
    Jan 28 at 9:42










  • $begingroup$
    I tried to differentiate it with respect to x
    $endgroup$
    – Kshitij Singh
    Jan 28 at 9:43










  • $begingroup$
    I tried to differentiate it with respect to x
    $endgroup$
    – Kshitij Singh
    Jan 28 at 9:43






  • 1




    $begingroup$
    Please typeset your equations using Mathjax for better presentation
    $endgroup$
    – Shubham Johri
    Jan 28 at 9:43








1




1




$begingroup$
What have you tried?
$endgroup$
– Parcly Taxel
Jan 28 at 9:42




$begingroup$
What have you tried?
$endgroup$
– Parcly Taxel
Jan 28 at 9:42












$begingroup$
I tried to differentiate it with respect to x
$endgroup$
– Kshitij Singh
Jan 28 at 9:43




$begingroup$
I tried to differentiate it with respect to x
$endgroup$
– Kshitij Singh
Jan 28 at 9:43












$begingroup$
I tried to differentiate it with respect to x
$endgroup$
– Kshitij Singh
Jan 28 at 9:43




$begingroup$
I tried to differentiate it with respect to x
$endgroup$
– Kshitij Singh
Jan 28 at 9:43




1




1




$begingroup$
Please typeset your equations using Mathjax for better presentation
$endgroup$
– Shubham Johri
Jan 28 at 9:43




$begingroup$
Please typeset your equations using Mathjax for better presentation
$endgroup$
– Shubham Johri
Jan 28 at 9:43










1 Answer
1






active

oldest

votes


















5












$begingroup$

Let $measuredangle ACB=90^{circ},$ $AC=3$, $BC=5$ and $CD$ be a bisector of $angle ACB$.



Also, let $CD=x$.



Thus, by the triangle inequality $$AD+BDgeq AB,$$ which gives $$a+b=sqrt{x^2+3^2-2xcdot3cdotcos45^{circ}}+sqrt{x^2+5^2-2xcdot5cdotcos45^{circ}}geqsqrt{3^2+5^2}=sqrt{34}.$$
The equality occurs, when $Din AB$, which says that we got a minimal value.



Now, by similarity we can show that $$CD^2=ACcdot BC-ADcdot BD$$ and since
$$frac{AD}{BD}=frac{AC}{BC}=frac{3}{5},$$ we obtain $$AD=frac{3}{8}sqrt{34},$$ $$BD=frac{5}{8}sqrt{34}$$ and
$$x=sqrt{3cdot5-frac{3}{8}sqrt{34}cdotfrac{5}{8}sqrt{34}}=frac{15}{4sqrt2}.$$






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%2f3090659%2fhow-to-find-minimum-value%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









    5












    $begingroup$

    Let $measuredangle ACB=90^{circ},$ $AC=3$, $BC=5$ and $CD$ be a bisector of $angle ACB$.



    Also, let $CD=x$.



    Thus, by the triangle inequality $$AD+BDgeq AB,$$ which gives $$a+b=sqrt{x^2+3^2-2xcdot3cdotcos45^{circ}}+sqrt{x^2+5^2-2xcdot5cdotcos45^{circ}}geqsqrt{3^2+5^2}=sqrt{34}.$$
    The equality occurs, when $Din AB$, which says that we got a minimal value.



    Now, by similarity we can show that $$CD^2=ACcdot BC-ADcdot BD$$ and since
    $$frac{AD}{BD}=frac{AC}{BC}=frac{3}{5},$$ we obtain $$AD=frac{3}{8}sqrt{34},$$ $$BD=frac{5}{8}sqrt{34}$$ and
    $$x=sqrt{3cdot5-frac{3}{8}sqrt{34}cdotfrac{5}{8}sqrt{34}}=frac{15}{4sqrt2}.$$






    share|cite|improve this answer











    $endgroup$


















      5












      $begingroup$

      Let $measuredangle ACB=90^{circ},$ $AC=3$, $BC=5$ and $CD$ be a bisector of $angle ACB$.



      Also, let $CD=x$.



      Thus, by the triangle inequality $$AD+BDgeq AB,$$ which gives $$a+b=sqrt{x^2+3^2-2xcdot3cdotcos45^{circ}}+sqrt{x^2+5^2-2xcdot5cdotcos45^{circ}}geqsqrt{3^2+5^2}=sqrt{34}.$$
      The equality occurs, when $Din AB$, which says that we got a minimal value.



      Now, by similarity we can show that $$CD^2=ACcdot BC-ADcdot BD$$ and since
      $$frac{AD}{BD}=frac{AC}{BC}=frac{3}{5},$$ we obtain $$AD=frac{3}{8}sqrt{34},$$ $$BD=frac{5}{8}sqrt{34}$$ and
      $$x=sqrt{3cdot5-frac{3}{8}sqrt{34}cdotfrac{5}{8}sqrt{34}}=frac{15}{4sqrt2}.$$






      share|cite|improve this answer











      $endgroup$
















        5












        5








        5





        $begingroup$

        Let $measuredangle ACB=90^{circ},$ $AC=3$, $BC=5$ and $CD$ be a bisector of $angle ACB$.



        Also, let $CD=x$.



        Thus, by the triangle inequality $$AD+BDgeq AB,$$ which gives $$a+b=sqrt{x^2+3^2-2xcdot3cdotcos45^{circ}}+sqrt{x^2+5^2-2xcdot5cdotcos45^{circ}}geqsqrt{3^2+5^2}=sqrt{34}.$$
        The equality occurs, when $Din AB$, which says that we got a minimal value.



        Now, by similarity we can show that $$CD^2=ACcdot BC-ADcdot BD$$ and since
        $$frac{AD}{BD}=frac{AC}{BC}=frac{3}{5},$$ we obtain $$AD=frac{3}{8}sqrt{34},$$ $$BD=frac{5}{8}sqrt{34}$$ and
        $$x=sqrt{3cdot5-frac{3}{8}sqrt{34}cdotfrac{5}{8}sqrt{34}}=frac{15}{4sqrt2}.$$






        share|cite|improve this answer











        $endgroup$



        Let $measuredangle ACB=90^{circ},$ $AC=3$, $BC=5$ and $CD$ be a bisector of $angle ACB$.



        Also, let $CD=x$.



        Thus, by the triangle inequality $$AD+BDgeq AB,$$ which gives $$a+b=sqrt{x^2+3^2-2xcdot3cdotcos45^{circ}}+sqrt{x^2+5^2-2xcdot5cdotcos45^{circ}}geqsqrt{3^2+5^2}=sqrt{34}.$$
        The equality occurs, when $Din AB$, which says that we got a minimal value.



        Now, by similarity we can show that $$CD^2=ACcdot BC-ADcdot BD$$ and since
        $$frac{AD}{BD}=frac{AC}{BC}=frac{3}{5},$$ we obtain $$AD=frac{3}{8}sqrt{34},$$ $$BD=frac{5}{8}sqrt{34}$$ and
        $$x=sqrt{3cdot5-frac{3}{8}sqrt{34}cdotfrac{5}{8}sqrt{34}}=frac{15}{4sqrt2}.$$







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Jan 28 at 10:11

























        answered Jan 28 at 9:45









        Michael RozenbergMichael Rozenberg

        109k1896201




        109k1896201






























            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%2f3090659%2fhow-to-find-minimum-value%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

            Npm cannot find a required file even through it is in the searched directory

            Image
            up vote 0 down vote favorite I got that error: users-Air:pdfuploader user$ npm run start client@0.1.0 start /Users/user/Documents/pdfuploader react-scripts start Could not find a required file. Name: index.html Searched in: /Users/user/Documents/pdfuploader/public npm ERR! code ELIFECYCLE npm ERR! errno 1 npm ERR! client@0.1.0 start: `react-scripts start` npm ERR! Exit status 1 npm ERR! npm ERR! Failed at the client@0.1.0 start script. npm ERR! This is probably not a problem with npm. There is likely additional logging output above. npm ERR! A complete log of this run can be found in: npm ERR! /Users/user/.npm/_logs/2018-11-19T10_19_14_555Z-debug.log users-Air:pdfuploader user$ And the file is in /Users/user/Documents/pdfuploader/public as you can see in my github repository: https://github.c...
            Read more