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# Find a function with continuous second derivative satisfying given conditions

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

1. for all , , and .
2. for all , , and .
3. for all , , and for all .
4. for all , , and for all .

1. 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.
2. Let . Then Furthermore, for all .

3. 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 .

4. 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 