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Find the absolute maximum and minimum values of f (, y) = { (73 – 1 – y2) + 3 on the region R= {(z,y) : 22 + y2 < 1} (the closed disk centered at (0, 0) with radius 1). = : SOLUTION We begin by locating the critical points of f on the interior of R. The critical points satisfy the equations f1(x, y) = = (3x2 - 1) = 0 and fy (2, y) = -y=0, which have the solutions 2 = + 1 and y = 0. The values of the function at these points are 5 (3,0) =3 - 37 and f ( 3,0) = 3 + alati = = We now determine the maximum and minimum values off on the boundary of R, which is a circle of radius 1 described by the parametric equations I = cos , y=sin 0, for O<O<27. Substituting 2 and y in terms of into the function f, we obtain a new function g(0) that gives the values of f on the boundary of R: 1 9(e) = } (cos* 0 – cos 0 – sin?0) +3. e =