2.2 Evaluate these partial derivatives at x = 1, y = 2.

Write the total differential for U.

Calculate dy/dx for dU = 0 — that is, what is the implied trade-off between x and y holding U constant?

Show U = 22 when x = 1, y = 2.

Suppose a firm’s total revenues depend on the amount produced (q) according to the function R =90q − q2 Total costs also depend on q:

C =q2 + 50q + 100 a.

What level of output should the irm produce to maximise proits (R – C)? What will proits be?

b.

c.

Show that the second-order conditions for a maximum are satisied at the output level found in part (a).

Does the solution calculated here obey the ‘marginal revenue equals marginal cost’ rule? Explain.