1. . A shipping company ships small crates for heavy materials that are secured by placing metal strapping around each of the three directions possible. Each band must have an overlap of 2 inches for the crimping device to tighten up the band. If the total length of metal strapping is 222 inches, find the dimensions(L,W, and H) of a box with maximum volume the shipping company can offer. Also, find the maximum volume.

2. A manufacturing company produces two products which are sold in two separate markets. The company's economists analyze the two markets and determine that the quantities q₁ and q₂, demanded by consumers and the prices p₁ and p₂ of each item are given by p₁ = 600 0.3_q1 and p₂ = 500 0.2_q2 The companies total production costs are given by C = 16 + 1.2_q1 + 1.5_q2+ 0.2_q1q2

a) Find the Revenue Function

b) Find the Profit Function

c) How much of each product should they produce to maximize profits?

3. Let f(x, y) = y² 2x²y + 4x³ + 20x²

a) Find the first order partial derivatives of f(x, y)

b) Find all critical points

c) Find all second order partial derivatives

d) Determine if the critical points correspond to local minima, local maxima, or saddle points.