2. Consider the function f(x,y) = 16xy -x⁴ -2y², Find all relative extrema and identify any saddle point.

8. For the equation 4x3y = 3 +y3 — 6x2 +5xy2. the slope of the tangent line at the point (2,3) is =____.

9. For the function f(x) = x3 - 3x2 - 24x 十 27 on the interval [-1,5], the absolute (global) maximum value is =____

10. The area between two curves y = 1 - x and y= ex from x = -1 to x = 4 is =____

【已刪除】1. Determine the improper integral dx is convergent or divergent.

3. Find the volume under the surface f(x,y) =   and over the domain D which is the region hounded by curves y = x, x = 0 and y = 3.

【已刪除】1. A monopolist’s total cost (TC) and total revenue (TR) functions are assumed to respectively，where Q denotes the output quantity. The government is going to impose a tax of t dollars per unit quantity on the commodity produced by the firm. The government wants to maximize T=tQ, the total revenue it realizes from the tax. However, the government does not know what tax rate t will accomplish that end. Find the tax rate t that maximizes total tax revenue 丁 for the government and the equilibrium quantity Q for which the firm’s profit is maximized after taxation. (20 points)

【已刪除】 (a) 

【已刪除】(b) ，where A，a, b, a and p are parameters. The prices of the two goods，x and y，and income are denoted by Px, Py，and I， respectively. (20 points)

(a) Determine Robin Crusoe’s optimal production/consumption pattern. (10 points) 

Suppose a capital market opens, which allows Robin to borrow or lend with the April Crusoe on the other islands. The borrowing and lending rate is r = 0.2.(b) What is the present value of Robin’s wealth assuming he uses the optimal production plan? (5 points)