Shows that Proposition 4.1 holds for negative integers also. The quotient of two polynomials is differentiable. Suppose x is a maximizer. If f is a minimizer the same argument holds with the inequalities reversed. The Lagrange Mean Value Theorem.
R is differentiable.
Then f and g differ by a constant iff their derivatives are equal. Then f is constant iff h and g differ by a constant. A similar argument applies to show that if the second derivative is negative, we have a local maximizer. A solution to the differential equation is a function which satisfies all of the conditions. Whenever the function f is a polynomial the solution is easy.