Let be a function differentiable everywhere such that
- Prove that and conjecture and prove a similar formula for .
- Conjecture and prove a formula for in terms of for all positive integers .
- Prove that and compute
- Prove that there exists a constant such that
for all . Find the value of the constant .
- Proof. We can compute this using the functional equation:
Next, we conjecture
Proof. Again, we compute using the functional equation, and the above formula for ,
- We conjecture
Proof. The proof is by induction. We have already established the cases and (and the case is the trivial ). Assume then that the formula holds for some integer . Then we have
Thus, if the formula holds for , it also holds for . Hence, by induction it holds for all integers
- Proof. Using the functional equation
Then, since the derivative exists for all (by hypothesis) we know it must exist in particular at . Using the limit definition of derivative, and the facts that and we have
- Proof. Since the derivative must exist for all (by hypothesis) we know that the limit
must exist for all . Using the functional equation for we have
But then, from part (c) we know and from this exercise (Section 6.17, Exercise #38) we know
Therefore, we have