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# Prove some properties of a differentiable function satisfying a given functional equation

Let be a function differentiable everywhere such that 1. Prove that and conjecture and prove a similar formula for .
2. Conjecture and prove a formula for in terms of for all positive integers .
3. Prove that and compute 4. Prove that there exists a constant such that for all . Find the value of the constant .

1. 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 , 2. 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 3. 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 4. 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 