For a1 < a2 < b2 < b1 and c > 0 real numbers show that there exists
a smooth (bump) function f : R → R with the following property: f(x) = ...
- 0 if x ≤ a and x ≥ b
- c if a2 ≤ x ≤ b2
- monotonic increasing if a1 < x < a2
- monotonic decreasing if b2 < x < b1
The Attempt at a Solution
So I'm not exactly sure what the question is asking. Is it asking me to construct such a function?
- If so then would I go about it by saying that c could be any constant like "5" or "2/3" and then go and simply give an example of a monotonic increasing function (like y = x) for that interval then give an example of a monotonic decreasing function (like y = -x) for that particular interval?
Or perhaps is the question asking me to show that a function like this is smooth?
- In which case we would essentially need to show that its a C-∞ function (at least all the examples I've seen say you need to show indefinite differentiation potential rather than just derivatives existing up to a certain, sufficient, degree.) However it doesn't appear that way since f(x) (given the example functions I've thrown in) would be twice differentiable, at most.
+ Unless of course then the implication is that we need to find monotonic functions which are also C-∞ functions in order to ensure the indefinite differentiability of f(x).
Any help anyone can provide about how to proceed or my proposed methods is extremely appreciated, I have this problem due for tomorrow morning and I'm just not quite sure what exactly needs to be shown here. Thanks in advance everyone, hoping for a lively and insightful conversation :)