Suppose that f,g : [a,b] → R. Decide which of the following statements are true and which are false. Prove the true ones and provide counterexamples for the false ones.
a) If f and g are increasing on [a, b], then f + g is increasing on [a, b].
b) If f and g are increasing on [a, b], then fg is increasing on [a, b].
c) If f is differentiable on (a, b) and limx→a+ f(x) exists and is finite, then for each x ∈ (a, b) there is a c between a and x such that f(x) - f(a+) = f'(c)(x - a).
d) If f and g are differentiable on [a, b] and |fʹ(x)| < 1 < |gʹ(x) < for all x ∈ (a, b), then |f(x) - f(a)| < |g(x) - g(a)| for all x ∈ [a, b].