Question

# 1. Find the partial derivatives; fx and fy:

(a) f(x,y) = 3x^{3}y^{3} -

5x^{2}y + 2xy^{2 }- 4xy^{3}

(b) f(x,y) = -(2x - 7)(5x+3y)

(c) f(x,y) = 5x(2y^{2}- 3x)^{3}

(d)f(x,y) = 10x / 5x - 4y^{2}

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

(a) f(x,y) = -3x^{2}2xy+100y^{2}

(b) f(x,y) = -x^{4}- xy - y^{4}

(c) f(x,y) = -3x^{2}y + 5xy^{2}

(d) f(x,y) = x^{1/3}y^{2/3}

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 = -5x^{2} - y^{2} + 2xy + 6x + 2y + 7

(b) z = 4x^{2} + xy + 2y^{2}

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 = -2x^{2} + 4xy - 3y^{2} +10x - 14y - 3

(b) z = -x^{2}- y^{2}- 2x + 4y + 8

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