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to follow the comment from ↑ Bati:,
I would add that I believe that also proving global uniqueness for solution on interval
is necessary to give the numerical solution some weight, assuming some numerical method for ODE was used
in case of ODE with more solutions, numerical approximation of one of the solutions
can be not anywhere close to full picture
(convexity and minimum of the function is very easy in this problem,
I'm just getting curious about the requested maximum estimate)
hi ↑ stuart clark:
There is therefore
together with it says that the function has at least local minimum in .
Furthermore is defined for so there is on the domain of the function
(note that inequality is sharp, never equality)
so in each expressions and are either both strictly positive or both strictly negative.
As and given and its derivatives are continuous, stays always positive
and therefore is concave upwards = convex.
This fact also says that is strictly increasing, and together with this gives
and , which makes the minimum in global.
To the last unanswered question I was yet able to find out that by analytical approach
but this is for sure not maximum as it cannot be reached.