Prove that these estimates cannot be improved












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I proved that for a generic function $f:Isubset mathbb{R}tomathbb{R}$ that is differentiable two times in the interval $I$ (open or closed, it makes no difference), we always have:



$$sup_{xin I}|f'(x)|leq 2sqrt{sup_{xin I}|f(x)| cdot sup_{xin I}|f''(x)|}quad (1)$$



when $Ineq mathbb{R}$, and:



$$sup_{xin I}|f'(x)|leq sqrt{2cdotsup_{xin I}|f(x)| cdot sup_{xin I}|f''(x)|}quad (2)$$



when $I = mathbb{R}$.



I now have to show that these estimates cannot be improved, so, in other words, I have to find two functions for which equality hold in the two cases.



For the first case, $f(x)=2x^2-1$ defined in $I=[0,1]$ does the job.



For the second case, I cannot find a function $f: mathbb{R}tomathbb{R}$ that verifies (2).



Have you any solutions?



$$$$



Note: in all this speech, there is an implicit hypothesis: $sup_{xin I}|f''(x)| neq 0$.






real-analysis derivatives supremum-and-infimum upper-lower-bounds







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    0












    $begingroup$


    I proved that for a generic function $f:Isubset mathbb{R}tomathbb{R}$ that is differentiable two times in the interval $I$ (open or closed, it makes no difference), we always have:



    $$sup_{xin I}|f'(x)|leq 2sqrt{sup_{xin I}|f(x)| cdot sup_{xin I}|f''(x)|}quad (1)$$



    when $Ineq mathbb{R}$, and:



    $$sup_{xin I}|f'(x)|leq sqrt{2cdotsup_{xin I}|f(x)| cdot sup_{xin I}|f''(x)|}quad (2)$$



    when $I = mathbb{R}$.



    I now have to show that these estimates cannot be improved, so, in other words, I have to find two functions for which equality hold in the two cases.



    For the first case, $f(x)=2x^2-1$ defined in $I=[0,1]$ does the job.



    For the second case, I cannot find a function $f: mathbb{R}tomathbb{R}$ that verifies (2).



    Have you any solutions?



    $$$$



    Note: in all this speech, there is an implicit hypothesis: $sup_{xin I}|f''(x)| neq 0$.






    real-analysis derivatives supremum-and-infimum upper-lower-bounds







    share|cite|improve this question









    $endgroup$















      0












      0








      0





      $begingroup$


      I proved that for a generic function $f:Isubset mathbb{R}tomathbb{R}$ that is differentiable two times in the interval $I$ (open or closed, it makes no difference), we always have:



      $$sup_{xin I}|f'(x)|leq 2sqrt{sup_{xin I}|f(x)| cdot sup_{xin I}|f''(x)|}quad (1)$$



      when $Ineq mathbb{R}$, and:



      $$sup_{xin I}|f'(x)|leq sqrt{2cdotsup_{xin I}|f(x)| cdot sup_{xin I}|f''(x)|}quad (2)$$



      when $I = mathbb{R}$.



      I now have to show that these estimates cannot be improved, so, in other words, I have to find two functions for which equality hold in the two cases.



      For the first case, $f(x)=2x^2-1$ defined in $I=[0,1]$ does the job.



      For the second case, I cannot find a function $f: mathbb{R}tomathbb{R}$ that verifies (2).



      Have you any solutions?



      $$$$



      Note: in all this speech, there is an implicit hypothesis: $sup_{xin I}|f''(x)| neq 0$.






      real-analysis derivatives supremum-and-infimum upper-lower-bounds







      share|cite|improve this question









      $endgroup$




      I proved that for a generic function $f:Isubset mathbb{R}tomathbb{R}$ that is differentiable two times in the interval $I$ (open or closed, it makes no difference), we always have:



      $$sup_{xin I}|f'(x)|leq 2sqrt{sup_{xin I}|f(x)| cdot sup_{xin I}|f''(x)|}quad (1)$$



      when $Ineq mathbb{R}$, and:



      $$sup_{xin I}|f'(x)|leq sqrt{2cdotsup_{xin I}|f(x)| cdot sup_{xin I}|f''(x)|}quad (2)$$



      when $I = mathbb{R}$.



      I now have to show that these estimates cannot be improved, so, in other words, I have to find two functions for which equality hold in the two cases.



      For the first case, $f(x)=2x^2-1$ defined in $I=[0,1]$ does the job.



      For the second case, I cannot find a function $f: mathbb{R}tomathbb{R}$ that verifies (2).



      Have you any solutions?



      $$$$



      Note: in all this speech, there is an implicit hypothesis: $sup_{xin I}|f''(x)| neq 0$.






      real-analysis derivatives supremum-and-infimum upper-lower-bounds




      real-analysis derivatives supremum-and-infimum upper-lower-bounds






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Jan 25 at 14:50









      NamelessNameless

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