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Find the absolute maximum and minimum values of f (,y) = (73 – 1 - y2) + 3 on the region R= {(x,y) : 22 + y2 <1} (the closed disk centered at (0, 0) with radius 1). T- = : SOLUTION We begin by locating the critical points of f on the interior of R. The critical points satisfy the equations f: (x, y) = - (3x2 - 1) = 0 and 2 fy (x, y) = -y=0, = which have the solutions x = + and y = 0. The values of the function at these points are f ( 153,0) = 3 - Bha and $ ( *;o) **,= 3 + 33 V3 33 We now determine the maximum and minimum values of f on the boundary of R, which is a circle of radius 1 described by the parametric equations <= cos , y=sin , for 0<0 < 27. Substituting 2 and y in terms of O into the function f, we obtain a new function g(0) that gives the values of f on the boundary of R: 1 g[ Ꮎ 9(0) = + (cos° 0 – cos 0 – sin) +3. 0 + . =