(a) Find the density for the standard normal random variable Z.

(b) Find f(z). Show that the only critical point for f occurs at z = 0. Use the first derivative test to show that f assumes its maximum value at z = 0.

(c) Find f"(z). Show that the possible inflection points occur at z = ±1. Use the second derivative to show that f changes concavity at z = +1 implying that the inflection points do occur when z = ± 1.

(d) Let X be normal with parameters p and a. Let (X - it.)/a = Z. Use the results of parts (b) and (c) to verify that, in general, a normal curve assumes its maximum value at x = µ and has points of inflection at x = µ ± a.