Question

5. Let f(x,y) = 3x2 y - y3 - 6x. (a) Find all the critical points off. (b) Classify each of the critical points; that is, what type are they? (c) For the same function f(x,y), find the maximum value of f on the unit square, 0 SX S1,0 <y s 1.

Answer 1

pointe, 02 2 5) frag). 30r 24-43-6 a) find all coitial step find fx & py fox & [3x²7-43-601) 6xy - 6 fy( Bay- y3.63 ) 2 [ 3x²y-y3. 60c] = 63 x ²-3 y Stec2find box & fyy & fay Par- & [6xy-6) = by foy - I [3x? 3)2) = -by & (6xy-6) oy Step 3; find point & (6xy. 6) = 0 392-3y?.0 x ² = ² . x y = 1 x=ty on og fxy= = 6a When faro & lo o & je = y2 Y = Il or when Y = 0 = 1 =-1 x=-1 are Critical points (11) & (-1,-1)

> Points 2 (a,b) points. 6) Classify each of the cortical step. I find discoiminant CD) at criticed Dca,y) = fxa carb) fcyy, ay) - CPayra, uj? -} for (x, y) = (1,1) faxa 661) = 6 Pyy="6(1)=-6 ; fxy = 6(1) = 6 : DC1, 1) =(6)-6) - (692 = -36-36 DC1,1) is co : it is saddle point for ca,y) = (-1,-1) fxx = 6(-1)=-6, fyy=-661): 6; Fay, 64-1)=-6 D(-1,-1) = (-6) (6) -E63² = -36-36 D(-1,-1) 40 : it is saddle points ) - 72 -) =-72 Both C1, 1) 4 (-1,-1) is saddle point. maxim on c] find Coo OCD1%!, OS K1 & on this region (1,1) there is one caitical point of (1,1). C1,0) we will find value of Pray) on (0,0); (0,1); (1, 1) 8 (1,0) & then Will maximan vaks of locy). decide

- maxim (100), f(1,0) 6 at cool fco,oo = fco,o) = 0-0-0 = aut -0-0-6 aut (1,1), f (1,1) 3-1-6 fro, 1)- 0 - 1-0 oll coll ill i 1 at (onl) i maxim value of ß on the remit sanas