If f : R R is such that f(x) = f(x) for all x R,

If f : R → R is such that fʹʹ(x) = f(x) for all x ∈ R, show that there exist real numbers a, b such that f(x) = ac(x) + bs(x) for all x ∈ R. Apply this to the functions f1(x) := ex and f2(x) := e-x for x ∈ R. Show that c(x) = 1/2 (ex + e-x) and s(x) = 1/2 (ex - e-x) for x ∈ R.