Exercise Set 3
More derivatives

1. Consider the function u(x,y) = ln(xy).

a. Obtain u₁, u₂, u₁₁, u₁₂, u₂₁ and u₂₂.

b. Show that u₁₁u₂₂ > u₁₂u₂₁.

2. Consider the function u(x,y) = ln(x²) + 2xy.

a. Obtain u₁, u₂, u₁₁, u₁₂, u₂₁ and u₂₂.

b. Is u₁₁u₂₂ - u₁₂u₂₁ > 0?

3. Consider the function L(x,y,m) = xy + m(300-2x-3y).

a. Obtain L₁, L₂ and L₃.

b. Set the first-order derivatives equal to zero. Solve for x, y and m. Call the solutions x₀, y₀ and m₀.

c. Find L(x₀, y₀, m₀).

4. Consider the functions f(x,y) = 4x^2/3y^1/3, and g(x,y) = 600 - 4x - 10y.

a. Set up a new function L(x,y,m) such that

L(x,y,m) = f(x,y) + mg(x,y)

b. Obtain L₁, L₂ and L₃.

c. Set the first-order derivatives equal to zero. Solve for x, y and m. Call the solutions x₀, y₀ and m₀.

c. Find L(x₀, y₀, m₀).