1. Let f and g be differentiable on R. Suppose that f (0) = g (0) and that f ′(x) ≤ g ′(x) for all x ≥ 0. Show that f (x) ≤ g (x) for all x ≥ 0 2. Let f be differentiable on (a, b) and continuous on...

1. Let f and g be differentiable on R. Suppose that f (0) = g (0) and that f ′(x) ≤ g ′(x) for all x ≥ 0. Show that f (x) ≤ g (x) for all x ≥ 0

2. Let f be differentiable on (a, b) and continuous on [a, b]. Suppose that f (a) = f (b) = 0. Apply Rolle’s theorem to the function g (x) = e^–kx
f (x) to prove that for each k ∈ R there exists c ∈ (a, b) such that f ′(c) = kf (c).