Question

3. (28 points) Let f(x,y) = 2x3 - 6xy+3y- be a function defined on xy-plane. (a) (6 pnts) Find first and second partial derivatives of f. (b) (10 pnts ) Determine the local extreme points of f (max., min., saddle points) if there is any. (C) (12 pnts) Find the maximum and minimum values of f over the closed region bounded by the lines y = -x, y = 1 and y=r

Answer 1

Let f(x,y) = 2x² - 6xy + 3y 2. a) Taking partial derivatives of f(x, 4) w.sto. re ty be= 6x² - GY 122 fy - - 6x+6y fox frey fyy - 6 6 f = الامام Set for ao 4 ty 6x² - by za - 6x+6y=0 = x²-y zo x = 1 x?a x(x-1) = 0 Now, when x=0 when x=1 y = 1 Hence, frany) has critical points. (0,o) (1) fex fyy - fey ? 12ax 6 co)? 72x - 36 at (a,b) = (0,0) To forn (a,0) = = 7280 - 36 --36 co = (0,oo is a soddle point (a,b) =(1, 1) from (1, 1) = 12x1 = 12 70 and D 72x1 - 36. 3670 =) by second desi vative test foy) has local minimum at (x,y) = (1, 1). at

c) Hence, the minimum value of fait) is -1 foul- 2 x 1 - 6xlx lt 3x1 = ليامي Let the regiono bounded by y=x, y = -x + y = 1 Then f has minimum value at (xal) = (IL) at (0,0) fro, os=0 fel D = 28 () – 6(-1)x3x] 7 at (-1, 1), - - 2 + 6+3 Hence, fouy has has maximum eut (x, y) = (-1,1) The maximum value is