Determine whether the following statements are true and give an explanation or counterexample. Assume that f is defined on R. (a) If f(x) > 0 and f(x) < 0 on an interval I, 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.