Consider the function f(x, y) = x 2 + 3 y 2 + 2 y on the closed disk S : x 2 + y 2

Consider the functionf(x, y) =x
² + 3y
² + 2y

on the closed diskS:x
² +y
² <>

a) Find the critical points in the interior ofSusing the first and second derivative tests, and decide if they are local maxima, minima, or neither.

b) Find the maximum and minimum off(x, y) on the boundary ofSby using Lagrange multipliers on the boundary ofS. (i.e. optimizef(x, y) subject to
the constraintx
² +y
² = 1).


(c) Using the results of (a) and (b), conclude what the global maximum and minimum values off(x, y) are, and where they are attained.