Determine whether the following statements are true and give an explanation or counterexample.

a. If f'(x) > 0 and f"(x) < 0 on an interval, then f is increasing at a decreasing rate on the interval.

b. If f'(c) > 0 and f"(c) = 0, then f has a local maximum at c.

c. Two functions that differ by an additive constant both increase and decrease on the same intervals.

d. If f and g increase on an interval, then the product fg also increases on that interval.

e. There exists a function f that is continuous on (-∞, ∞) with exactly three critical points, all of which correspond to local maxima.