In each of the following cases find a function with continuous second derivative satisfying the given conditions.

- for all , , and .
- for all , , and .
- for all , , and for all .
- for all , , and for all .

- There can be no function meeting all of these conditions since implies is an increasing function (since its derivative, , is positive). But then contradicts that is increasing.
- Let . Then
Furthermore, for all .

- There can be no function meeting all of these conditions. Again, for all implies that is increasing for all . Therefore, implies for all . Then, by the mean-value theorem, we know that for any there exists some such that
Now, choose . Then, , contradicting that for all .

- We’ll define piecewise as follows
Then, we can take the derivative of each piece (and see that they are equal, so the derivative is defined at )

Taking the derivative again we find

Thus, for all and . Furthermore, for we have