Mixing
Prove that these estimates cannot be improved
$begingroup$
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?
$$$$
Note: in all this speech, there is an implicit hypothesis: $sup_{xin I}|f''(x)| neq 0$.
real-analysis derivatives supremum-and-infimum upper-lower-bounds
asked Jan 25 at 14:50
NamelessNameless
5010
$endgroup$
add a comment |
$begingroup$
I proved that for a generic function $f:Isubset mathbb{R}tomathbb{R}$ that is differentiable two times in the interval $I$ (open or closed, it makes no difference), we always have:
$$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)$$
when $I = mathbb{R}$.
I now have to show that these estimates cannot be improved, so, in other words, I have to find two functions for which equality hold in the two cases.
For the first case, $f(x)=2x^2-1$ defined in $I=[0,1]$ does the job.
For the second case, I cannot find a function $f: mathbb{R}tomathbb{R}$ that verifies (2).
Have you any solutions?
$$$$
Note: in all this speech, there is an implicit hypothesis: $sup_{xin I}|f''(x)| neq 0$.
real-analysis derivatives supremum-and-infimum upper-lower-bounds
asked Jan 25 at 14:50
NamelessNameless
5010
$endgroup$
add a comment |
$begingroup$
I proved that for a generic function $f:Isubset mathbb{R}tomathbb{R}$ that is differentiable two times in the interval $I$ (open or closed, it makes no difference), we always have:
$$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)$$
when $I = mathbb{R}$.
I now have to show that these estimates cannot be improved, so, in other words, I have to find two functions for which equality hold in the two cases.
For the first case, $f(x)=2x^2-1$ defined in $I=[0,1]$ does the job.
For the second case, I cannot find a function $f: mathbb{R}tomathbb{R}$ that verifies (2).
Have you any solutions?
$$$$
Note: in all this speech, there is an implicit hypothesis: $sup_{xin I}|f''(x)| neq 0$.
real-analysis derivatives supremum-and-infimum upper-lower-bounds
asked Jan 25 at 14:50
NamelessNameless
5010
$endgroup$
I proved that for a generic function $f:Isubset mathbb{R}tomathbb{R}$ that is differentiable two times in the interval $I$ (open or closed, it makes no difference), we always have:
$$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
asked Jan 25 at 14:50
NamelessNameless
5010
asked Jan 25 at 14:50
NamelessNameless
5010
asked Jan 25 at 14:50
NamelessNameless
5010
asked Jan 25 at 14:50
NamelessNameless
5010
asked Jan 25 at 14:50
NamelessNameless
5010
5010
add a comment |
add a comment |
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