State the mean value theorem and deduce it from Rolle's theorem.
Use the mean value theorem to show that, if is differentiable with for all , then is constant.
By considering the derivative of the function given by , find all the solutions of the differential equation where is differentiable and is a fixed real number.
Show that, if is continuous, then the function given by
is differentiable with .
Find the solution of the equation
where is differentiable and is a real number. You should explain why the solution is unique.