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1. Find the partial derivatives; fx and fy:


(a) f(x,y) = 3x3y3 -

5x2y + 2xy2 - 4xy3


(b) f(x,y) = -(2x - 7)(5x+3y)


(c) f(x,y) = 5x(2y2- 3x)3


(d)f(x,y) = 10x / 5x - 4y2



2. Find the second order partial derivatives; fxx, fyy and fxy (cross partial derivatives):


(a) f(x,y) = -3x22xy+100y2


(b) f(x,y) = -x4- xy - y4


(c) f(x,y) = -3x2y + 5xy2


(d) f(x,y) = x1/3y2/3



3. Show first that the following function is concave (a) and convex (b) so that we have max and min, respectively. Then find its global max/min:


(a) z = -5x2 - y2 + 2xy + 6x + 2y + 7


(b) z = 4x2 + xy + 2y2



4. Optimize the following functions by finding the extreme values at which the function is stationary and determine whether the points are (local) maxima or minima, or saddle points:


(a) z = -2x2 + 4xy - 3y2 +10x - 14y - 3


(b) z = -x2- y2- 2x + 4y + 8

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