Hi all,
Need some help with this simple? proof.
Let A be the set of all twice differentiable functions with the properties: , and .
Prove: If and if
for all , then with all .
Thanks.
suppose first that there exists such that then by the mean value theorem for some thus again, by the mean value theorem
for some but since is decreasing we have and hence i.e. thus so
now suppose for some then, since is continuous, there exists such that for all but then we'll have:
thus and hence contradiction! so there's no with and we're done.