Let f(x) = (a - x)^x, where a > 0. 

a. What is the domain of f (in terms of a)?

b. Describe the end behavior of f (near the boundary of its domain).

c. Compute f'. Then graph f and f' for a = 0.5, 1, 2, and 3.

d. Show that f has a single local maximum at the point z that satisfies z = (a - z) ln (a - z).

e. Describe how z (found in part (d)) varies as a increases. Describe how f(z) varies as a increases.