1. 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.

2.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 (