Consider the Cobb-Douglas production model for a manufacturing process depending on three inputs x, y, and z

Question:

Consider the Cobb-Douglas production model for a manufacturing process depending on three inputs x, y, and z with unit costs a, b, and c, respectively, given by
P = kx(y(z(, ( > 0, ( > 0, ( > 0, ( + ( + ( = 1
Subject to the cost constraint ax + by + a = 4. Determine x, y, and z to maximize the production P.