Question

2.10.4 Given a function f(x,y) on a compact region E in R^2, Find the maximum and minimum values of f on E, and the points at which these extreme values are attained.

f(x, y) = x2 sin y + x, and E is the filled rectangle where -1 < x < 1 and | 0 < a < .

Answer 1

Solution tably) = x sinytx. -Ilahl osyst For extremum, at at =0 = ) oy an x² cosy=0 - 2xsinyt =0 isiny=-Y2x xzo or losy zo 2 x=0 or y=1,38 tas y Cost] IT y=F/2. If y = 372 - 1 ---Y2x - 1=-1₂2 16=22 10=12 Stationen ting points (-72272) (2 37 ) Nord, 24 = asing azt = -x² sing ay2 227 = ax cosy Aut (-Y2 35h) At (Y21372) 24 = 20 2²7 = -2 co 24 = -² 24 0 227 =0 2x2 Bay ax² azt = Xy dyz axay duay

4 (2,384) B+ 》NA) 曼等) = 0, 22 2 pradeo Neither Maxima Nor Minima.