General statements for the second derivative of a function

The name of the pictureThe name of the pictureThe name of the pictureClash Royale CLAN TAG#URR8PPP











up vote
1
down vote

favorite












I am working on a task about second derivative. The task is:



$f(x)$ on $(-1,1)$ has the values $f(-1)=-10$, $f(0)=-10$ and $f(1)=-3$.



What can you say about the values for first and second derivative?



For the first derivative I use the mean value theorem and find $f'(c)$ for different intervals.



For the second derivative I have some statements:



1) $|f '' (c)|>frac72$



2) $|f '' (c)|>7$



Are there any theorems or rules I can use in order to check if these statements are true or not?



Thanks!










share|cite|improve this question



























    up vote
    1
    down vote

    favorite












    I am working on a task about second derivative. The task is:



    $f(x)$ on $(-1,1)$ has the values $f(-1)=-10$, $f(0)=-10$ and $f(1)=-3$.



    What can you say about the values for first and second derivative?



    For the first derivative I use the mean value theorem and find $f'(c)$ for different intervals.



    For the second derivative I have some statements:



    1) $|f '' (c)|>frac72$



    2) $|f '' (c)|>7$



    Are there any theorems or rules I can use in order to check if these statements are true or not?



    Thanks!










    share|cite|improve this question

























      up vote
      1
      down vote

      favorite









      up vote
      1
      down vote

      favorite











      I am working on a task about second derivative. The task is:



      $f(x)$ on $(-1,1)$ has the values $f(-1)=-10$, $f(0)=-10$ and $f(1)=-3$.



      What can you say about the values for first and second derivative?



      For the first derivative I use the mean value theorem and find $f'(c)$ for different intervals.



      For the second derivative I have some statements:



      1) $|f '' (c)|>frac72$



      2) $|f '' (c)|>7$



      Are there any theorems or rules I can use in order to check if these statements are true or not?



      Thanks!










      share|cite|improve this question















      I am working on a task about second derivative. The task is:



      $f(x)$ on $(-1,1)$ has the values $f(-1)=-10$, $f(0)=-10$ and $f(1)=-3$.



      What can you say about the values for first and second derivative?



      For the first derivative I use the mean value theorem and find $f'(c)$ for different intervals.



      For the second derivative I have some statements:



      1) $|f '' (c)|>frac72$



      2) $|f '' (c)|>7$



      Are there any theorems or rules I can use in order to check if these statements are true or not?



      Thanks!







      calculus functions






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Sep 9 at 13:51









      rogerl

      16.7k22745




      16.7k22745










      asked Sep 9 at 13:40









      netwon1227

      119110




      119110




















          1 Answer
          1






          active

          oldest

          votes

















          up vote
          0
          down vote













          Hint



          If we consider $xin(-1,1)$ then



          $$|f'(x)|le kimplies |f(x)-f(0)|le kleft|xright|le kimplies |f(1)-f(0)|le k.$$



          Since $f(1)-f(0)=7,$ what can we say about $k?$






          share|cite|improve this answer






















          • We can say that k>= 7 , but I don't see how it can say anything about f ''(x). Sorry.
            – netwon1227
            Sep 9 at 14:40











          • There exist $a,bin (-1,1)$ such that $f'(a)=0$ and $f'(b)>7.$ Use that $7<f'(b)-f'(a)=f''(c)(b-a).$ What do you get?
            – mfl
            Sep 9 at 15:05











          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: false,
          showLowRepImageUploadWarning: true,
          reputationToPostImages: 10,
          bindNavPrevention: true,
          postfix: "",
          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%2f2910796%2fgeneral-statements-for-the-second-derivative-of-a-function%23new-answer', 'question_page');

          );

          Post as a guest






























          1 Answer
          1






          active

          oldest

          votes








          1 Answer
          1






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes








          up vote
          0
          down vote













          Hint



          If we consider $xin(-1,1)$ then



          $$|f'(x)|le kimplies |f(x)-f(0)|le kleft|xright|le kimplies |f(1)-f(0)|le k.$$



          Since $f(1)-f(0)=7,$ what can we say about $k?$






          share|cite|improve this answer






















          • We can say that k>= 7 , but I don't see how it can say anything about f ''(x). Sorry.
            – netwon1227
            Sep 9 at 14:40











          • There exist $a,bin (-1,1)$ such that $f'(a)=0$ and $f'(b)>7.$ Use that $7<f'(b)-f'(a)=f''(c)(b-a).$ What do you get?
            – mfl
            Sep 9 at 15:05















          up vote
          0
          down vote













          Hint



          If we consider $xin(-1,1)$ then



          $$|f'(x)|le kimplies |f(x)-f(0)|le kleft|xright|le kimplies |f(1)-f(0)|le k.$$



          Since $f(1)-f(0)=7,$ what can we say about $k?$






          share|cite|improve this answer






















          • We can say that k>= 7 , but I don't see how it can say anything about f ''(x). Sorry.
            – netwon1227
            Sep 9 at 14:40











          • There exist $a,bin (-1,1)$ such that $f'(a)=0$ and $f'(b)>7.$ Use that $7<f'(b)-f'(a)=f''(c)(b-a).$ What do you get?
            – mfl
            Sep 9 at 15:05













          up vote
          0
          down vote










          up vote
          0
          down vote









          Hint



          If we consider $xin(-1,1)$ then



          $$|f'(x)|le kimplies |f(x)-f(0)|le kleft|xright|le kimplies |f(1)-f(0)|le k.$$



          Since $f(1)-f(0)=7,$ what can we say about $k?$






          share|cite|improve this answer














          Hint



          If we consider $xin(-1,1)$ then



          $$|f'(x)|le kimplies |f(x)-f(0)|le kleft|xright|le kimplies |f(1)-f(0)|le k.$$



          Since $f(1)-f(0)=7,$ what can we say about $k?$







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Sep 9 at 14:02

























          answered Sep 9 at 13:56









          mfl

          25.2k12141




          25.2k12141











          • We can say that k>= 7 , but I don't see how it can say anything about f ''(x). Sorry.
            – netwon1227
            Sep 9 at 14:40











          • There exist $a,bin (-1,1)$ such that $f'(a)=0$ and $f'(b)>7.$ Use that $7<f'(b)-f'(a)=f''(c)(b-a).$ What do you get?
            – mfl
            Sep 9 at 15:05

















          • We can say that k>= 7 , but I don't see how it can say anything about f ''(x). Sorry.
            – netwon1227
            Sep 9 at 14:40











          • There exist $a,bin (-1,1)$ such that $f'(a)=0$ and $f'(b)>7.$ Use that $7<f'(b)-f'(a)=f''(c)(b-a).$ What do you get?
            – mfl
            Sep 9 at 15:05
















          We can say that k>= 7 , but I don't see how it can say anything about f ''(x). Sorry.
          – netwon1227
          Sep 9 at 14:40





          We can say that k>= 7 , but I don't see how it can say anything about f ''(x). Sorry.
          – netwon1227
          Sep 9 at 14:40













          There exist $a,bin (-1,1)$ such that $f'(a)=0$ and $f'(b)>7.$ Use that $7<f'(b)-f'(a)=f''(c)(b-a).$ What do you get?
          – mfl
          Sep 9 at 15:05





          There exist $a,bin (-1,1)$ such that $f'(a)=0$ and $f'(b)>7.$ Use that $7<f'(b)-f'(a)=f''(c)(b-a).$ What do you get?
          – mfl
          Sep 9 at 15:05


















           

          draft saved


          draft discarded















































           


          draft saved


          draft discarded














          StackExchange.ready(
          function ()
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2910796%2fgeneral-statements-for-the-second-derivative-of-a-function%23new-answer', 'question_page');

          );

          Post as a guest













































































          這個網誌中的熱門文章

          How to combine Bézier curves to a surface?

          Why am i infinitely getting the same tweet with the Twitter Search API?

          Carbon dioxide