The adult bookstores raise three issues on appeal. First, they argue that neither preclusion nor theRooker-Feldmandoctrine bar their challenge to Broward County's licensing ordinance. Second, they question the district court's ruling that Broward County's zoning ordinance is facially constitutional. Finally, appellants argue that, as applied, the zoning ordinance violates the First Amendment because it denies adequate opportunities for adult expression.

We agree with appellants that their prior efforts to obtain a temporary injunction of Broward County's licensing ordinance does not bar a subsequent claim for a permanent injunction. This circuit's precedent does, however, support the district court's ruling that the zoning ordinance is facially constitutional. Furthermore, we cannot say that the district court's findings as to the number of sites available for adult businesses under the zoning ordinance are clearly erroneous, and we agree that those sites provide an adequate opportunity for the appellants' protected expression. We reverse the order precluding appellants' challenge to Broward County's licensing ordinance and remand for further proceedings. We affirm the district court ruling that Broward County's zoning ordinance is constitutional both facially and as applied.

1. Differentiate the following functions:

(a) f(x) = 6 + + .

2. Determine whether the following functions are strictly convex, strictly concave, or neither over the specified intervals:

(a)() = + , for x = any real number.

(b)() = , for x > 0.

(c)() = , for x .

(d)() = + , for x 0.

3.Find the values of x₁ and x₂ which maximize

(,) = + + .

4.Let f(x₁, x₂) = A, where A, , > 0, be defined for the domain x₁, x₂ > 0. Demonstrate that the function is strictly concave within its domain if and only if + < 1.

5.Find the values for x₁ and x₂ that maximize f (x₁, x₂) = subject to the requirement that 5x₁ + 2x₂ = 300. Demonstrate that the appropriate second-order condition is satisfied.

6. Find functions of two variables with the domains x₁, x₂ > 0 that are (a) Quasi-concave, but not strictly quasi-concave and not concave.

(b)Strictly quasi-concave, but not concave.

(c)Quasi-concave, but not strictly quasi-concave and not strictly concave.

(d)Strictly quasi-concave and concave, but not strictly concave.

7.The locus of points of tangency between income lines and indifference curves for given prices p₁, p₂ and a changing value of income is called an income expansion line or Engel curve. Show that the Engel curve is a straight line if the utility function is given byU = _, > .

8.Let a consumer's utility function be U = + . + and his budget constraint 3 + = . Show that his optimum commodity bundle is the same as in Exercise 2.3. Why is this the case?

9.Prove that if the consumer is indifferent between commodity bundles ( and and has a homothetic utility function, she will also be indifferent between the

bundles () and (

10.Construct an indirect utility function that corresponds to the direct function U =

+ . Use Roy's identity to construct demand functions for the two goods. Are these the same as the demand functions derived from the direct utility function?