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 left boundary of its domain and as x→∞).

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

d. Show that f has a single local minimum at the point z that satisfies (z + a) ln (z + a) + z = 0.

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