1. Calculate:

(a) partial derivative symbol[4x^2 - 3xy + y^2]/partial derivative x. Then, in addition, take the partial derivative of your answer with respect to y.

This is called a cross-partial derivative.

(b) partial derivative symbol[4x^2 - 3xy + y^2]/partial derivative symbol y. Then, in addition, take the partial derivative of your answer with respect to x.

Compare this cross-partial derivative to the one that you calculated in part (a).

(c) partial derivative symbol^2[4x^2 - 3xy + y^2]/part derivative x^2.   (*all the M are partial derivative symbol, the computer did not recognized the sign when i copied and paste.

(d) M2[4x^2 - 3xy + y^2]/My^2.

(e) M2[4x^2ln(xy)]/Mx^2 (Simplify your answer as much as possible.)

2. Assume that the demand for welding services is D(P) = 34-P/2, and the inverse supply function is PS(Q)= 4 + 10Q + Q2. This supply curve represents the long run supply curve of the incumbent firms already in the market, but there is free entry into the market.

(a) At what market prices can a new firm enter profitably?

The last three parts refer to a long run that includes the actions of potential entrants.

(b) What is the long run equilibrium price?

(c) Draw the long run supply curve, and calculate the inverse long run supply function.

(d) What is the long run equilibrium quantity?

3. The problem is to choose x to maximize f(x;alpha) = alpha x - 3x^2, where alpha is exogenous, subject to the constraint x$0.

(a) For this problem, which values of x are on the boundary, and which are in the interior?

(b) Calculate the first-order condition for this problem.

(c) If alpha=12, then which value of x solves the problem? Justify your answer carefully.

(d) If alpha=-2, then which value of x solves the problem? Justify your answer carefully. What is the maximized value of f?

(e) For which possible values of alpha does this problem have a solution on the boundary?