What is the value of of the expression below












0














If $f(0) =0$ and $f''(x)$ exists then what is the value of $$f'(x)-frac{f(x)}{x}$$



I tried Tailor and Lagrange theorems but to no avail.



clearly $f''(c)=2frac{f(x)-xf'(0
)}{x^2}$ where $0<c<x$
but how can i proceed further.
How can i find $f'(0)$






calculus







share|cite|improve this question
























  • You can't, it's totally arbitrary.
    – Yves Daoust
    Mar 15 '16 at 10:13


















0














If $f(0) =0$ and $f''(x)$ exists then what is the value of $$f'(x)-frac{f(x)}{x}$$



I tried Tailor and Lagrange theorems but to no avail.



clearly $f''(c)=2frac{f(x)-xf'(0
)}{x^2}$ where $0<c<x$
but how can i proceed further.
How can i find $f'(0)$






calculus







share|cite|improve this question
























  • You can't, it's totally arbitrary.
    – Yves Daoust
    Mar 15 '16 at 10:13
















0












0








0







If $f(0) =0$ and $f''(x)$ exists then what is the value of $$f'(x)-frac{f(x)}{x}$$



I tried Tailor and Lagrange theorems but to no avail.



clearly $f''(c)=2frac{f(x)-xf'(0
)}{x^2}$ where $0<c<x$
but how can i proceed further.
How can i find $f'(0)$






calculus







share|cite|improve this question















If $f(0) =0$ and $f''(x)$ exists then what is the value of $$f'(x)-frac{f(x)}{x}$$



I tried Tailor and Lagrange theorems but to no avail.



clearly $f''(c)=2frac{f(x)-xf'(0
)}{x^2}$ where $0<c<x$
but how can i proceed further.
How can i find $f'(0)$






calculus




calculus






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Mar 15 '16 at 8:41

























asked Mar 15 '16 at 8:21









Rayees Ahmad

884413




884413












  • You can't, it's totally arbitrary.
    – Yves Daoust
    Mar 15 '16 at 10:13




















  • You can't, it's totally arbitrary.
    – Yves Daoust
    Mar 15 '16 at 10:13


















You can't, it's totally arbitrary.
– Yves Daoust
Mar 15 '16 at 10:13






You can't, it's totally arbitrary.
– Yves Daoust
Mar 15 '16 at 10:13












3 Answers
3






active

oldest

votes


















1














Expanding $f(t)$ around $x$ gives $f(t)=f(x)+(t-x)f'(x)+frac{(t-x)^2}2f''(tau)$, where $tau$ lies between $t$ and $x$.



For $t=0$ this gives $0=f(0)=f(x)-xf'(x)+frac{x^2}2f''(theta x)$ with $theta in (0,1)$ and therefore $f'(x)-frac{f(x)}x=frac x2f''(theta x)$.






share|cite|improve this answer































    0














    It is not clear what you are asking, though there are many functions that satisfy your conditions but yielding very different expressions.


    For example, take $f(x)=x$ for which $f(0)=0, f''(x)$ exists and equals $0$.
    That gives you an expression for $$f'(x)-frac{f(x)}{x}=1-1=0$$


    and then take $g(x)=sinx$, $g(0)=0,g''(x)=-sinx$ for which

    $$g'(x)-frac{g(x)}{x}=cosx-frac{sinx}{x}$$

    Two different "values" for two functions that satisfy your conditions.






    share|cite|improve this answer





























      0














      By Taylor,



      $$f'(x)-frac{f(x)}x=frac x2f''(0)+r'(x)-frac{r(x)}x,$$ where $r(x)$ is the remainder (hopefully, the last two terms can be neglectible for small $x$).






      share|cite|improve this answer





















        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%2f1698318%2fwhat-is-the-value-of-of-the-expression-below%23new-answer', 'question_page');
        }
        );

        Post as a guest















        Required, but never shown

























        3 Answers
        3






        active

        oldest

        votes








        3 Answers
        3






        active

        oldest

        votes









        active

        oldest

        votes






        active

        oldest

        votes









        1














        Expanding $f(t)$ around $x$ gives $f(t)=f(x)+(t-x)f'(x)+frac{(t-x)^2}2f''(tau)$, where $tau$ lies between $t$ and $x$.



        For $t=0$ this gives $0=f(0)=f(x)-xf'(x)+frac{x^2}2f''(theta x)$ with $theta in (0,1)$ and therefore $f'(x)-frac{f(x)}x=frac x2f''(theta x)$.






        share|cite|improve this answer




























          1














          Expanding $f(t)$ around $x$ gives $f(t)=f(x)+(t-x)f'(x)+frac{(t-x)^2}2f''(tau)$, where $tau$ lies between $t$ and $x$.



          For $t=0$ this gives $0=f(0)=f(x)-xf'(x)+frac{x^2}2f''(theta x)$ with $theta in (0,1)$ and therefore $f'(x)-frac{f(x)}x=frac x2f''(theta x)$.






          share|cite|improve this answer


























            1












            1








            1






            Expanding $f(t)$ around $x$ gives $f(t)=f(x)+(t-x)f'(x)+frac{(t-x)^2}2f''(tau)$, where $tau$ lies between $t$ and $x$.



            For $t=0$ this gives $0=f(0)=f(x)-xf'(x)+frac{x^2}2f''(theta x)$ with $theta in (0,1)$ and therefore $f'(x)-frac{f(x)}x=frac x2f''(theta x)$.






            share|cite|improve this answer














            Expanding $f(t)$ around $x$ gives $f(t)=f(x)+(t-x)f'(x)+frac{(t-x)^2}2f''(tau)$, where $tau$ lies between $t$ and $x$.



            For $t=0$ this gives $0=f(0)=f(x)-xf'(x)+frac{x^2}2f''(theta x)$ with $theta in (0,1)$ and therefore $f'(x)-frac{f(x)}x=frac x2f''(theta x)$.







            share|cite|improve this answer














            share|cite|improve this answer



            share|cite|improve this answer








            edited Sep 19 at 23:48

























            answered Sep 19 at 23:39









            random

            47126




            47126























                0














                It is not clear what you are asking, though there are many functions that satisfy your conditions but yielding very different expressions.


                For example, take $f(x)=x$ for which $f(0)=0, f''(x)$ exists and equals $0$.
                That gives you an expression for $$f'(x)-frac{f(x)}{x}=1-1=0$$


                and then take $g(x)=sinx$, $g(0)=0,g''(x)=-sinx$ for which

                $$g'(x)-frac{g(x)}{x}=cosx-frac{sinx}{x}$$

                Two different "values" for two functions that satisfy your conditions.






                share|cite|improve this answer


























                  0














                  It is not clear what you are asking, though there are many functions that satisfy your conditions but yielding very different expressions.


                  For example, take $f(x)=x$ for which $f(0)=0, f''(x)$ exists and equals $0$.
                  That gives you an expression for $$f'(x)-frac{f(x)}{x}=1-1=0$$


                  and then take $g(x)=sinx$, $g(0)=0,g''(x)=-sinx$ for which

                  $$g'(x)-frac{g(x)}{x}=cosx-frac{sinx}{x}$$

                  Two different "values" for two functions that satisfy your conditions.






                  share|cite|improve this answer
























                    0












                    0








                    0






                    It is not clear what you are asking, though there are many functions that satisfy your conditions but yielding very different expressions.


                    For example, take $f(x)=x$ for which $f(0)=0, f''(x)$ exists and equals $0$.
                    That gives you an expression for $$f'(x)-frac{f(x)}{x}=1-1=0$$


                    and then take $g(x)=sinx$, $g(0)=0,g''(x)=-sinx$ for which

                    $$g'(x)-frac{g(x)}{x}=cosx-frac{sinx}{x}$$

                    Two different "values" for two functions that satisfy your conditions.






                    share|cite|improve this answer












                    It is not clear what you are asking, though there are many functions that satisfy your conditions but yielding very different expressions.


                    For example, take $f(x)=x$ for which $f(0)=0, f''(x)$ exists and equals $0$.
                    That gives you an expression for $$f'(x)-frac{f(x)}{x}=1-1=0$$


                    and then take $g(x)=sinx$, $g(0)=0,g''(x)=-sinx$ for which

                    $$g'(x)-frac{g(x)}{x}=cosx-frac{sinx}{x}$$

                    Two different "values" for two functions that satisfy your conditions.







                    share|cite|improve this answer












                    share|cite|improve this answer



                    share|cite|improve this answer










                    answered Mar 15 '16 at 10:03









                    MathematicianByMistake

                    3,5831726




                    3,5831726























                        0














                        By Taylor,



                        $$f'(x)-frac{f(x)}x=frac x2f''(0)+r'(x)-frac{r(x)}x,$$ where $r(x)$ is the remainder (hopefully, the last two terms can be neglectible for small $x$).






                        share|cite|improve this answer


























                          0














                          By Taylor,



                          $$f'(x)-frac{f(x)}x=frac x2f''(0)+r'(x)-frac{r(x)}x,$$ where $r(x)$ is the remainder (hopefully, the last two terms can be neglectible for small $x$).






                          share|cite|improve this answer
























                            0












                            0








                            0






                            By Taylor,



                            $$f'(x)-frac{f(x)}x=frac x2f''(0)+r'(x)-frac{r(x)}x,$$ where $r(x)$ is the remainder (hopefully, the last two terms can be neglectible for small $x$).






                            share|cite|improve this answer












                            By Taylor,



                            $$f'(x)-frac{f(x)}x=frac x2f''(0)+r'(x)-frac{r(x)}x,$$ where $r(x)$ is the remainder (hopefully, the last two terms can be neglectible for small $x$).







                            share|cite|improve this answer












                            share|cite|improve this answer



                            share|cite|improve this answer










                            answered Mar 15 '16 at 10:20









                            Yves Daoust

                            124k671221




                            124k671221






























                                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.





                                Some of your past answers have not been well-received, and you're in danger of being blocked from answering.


                                Please pay close attention to the following guidance:


                                • 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.


                                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%2f1698318%2fwhat-is-the-value-of-of-the-expression-below%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

                                How to send String Array data to Server using php in android

                                Image
                                0 I have a String array which contains value string array = {6,37,38,837,7,...} and I have also used some string values. I want to send that all string values and string array to my database which is located at server side. When I am click on submit button the data will stored in table. On the PHP side I want to read this array as it is,other thing i will do at background means PHP side. See my below code then you will get it what I am says. Here is my code: public class Hotels extends Activity { // Declare Variables JSONArray jsonarray = null; private static final String TAG_ID = "id"; public static final String TAG_NAME = "name"; public static final String TAG_LOCATION = "location"; public static final String TAG_DESC = "description"; String f_date, l_date;...
                                Read more

                                Title Spacing in Bjornstrup Chapter, Removing Chapter Number From Contents

                                Image
                                up vote 5 down vote favorite 2 I would like to make it so that the vertical space below the title of each chapter is the same when using the Bjornstrup package (i.e. so that the descenders do not affect the spacing - as shown by the red arrow in the picture below). I think I'm fairly happy with the spacing around "A chapter", so I'd really like titles without descenders (e.g. "Contents") to have the same vertical spacing as "A chapter" below the title. I would also like to remove the large "0" in the contents title. Any help appreciated, as always. documentclass[11pt,a4paper]{book} % Packages[![enter image description here][1]][1] usepackage[left=3.5cm, right=3.5cm, top=4.3cm, bottom=4.25cm]{geometry} usepackage[utf8]{inputenc} usepackage{libertine} usepackage[libertine]{newtx...
                                Read more

                                Is anime1.com a legal site for watching anime?

                                Image
                                up vote 4 down vote favorite 1 Other anime streaming sites are blocking my IP because I'm Iranian while anime1.com isn't. I wanted to know if it's legal to watch/stream anime from this site. Thanks. anime-production share | improve this question edited Nov 27 at 22:38 W. Are 1,599 2 23 asked Nov 27 at 20:36 mhnajafi 23 3 add a comment ...
                                Read more