General statements for the second derivative of a function

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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







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    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!






    calculus functions







    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!






      calculus functions







      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




      calculus functions






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      share|cite|improve this question













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      edited Sep 9 at 13:51









      rogerl

      16.7k22745




      16.7k22745










      asked Sep 9 at 13:40









      netwon1227

      119110




      119110




















          1 Answer
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          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




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          1 Answer
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          active

          oldest

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          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


















           

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