MATH 16B Final: finalreviewanswers.pdf
Document Summary
Let f(x, y) = 3x2 + 6xy + 4y2 + 4y. Find all maxima, minima, and saddle points of f(x, y). At a maximum, minimum, or saddle point, the partial derivatives f have. We solve these equations for x and y. Substituting for y in (2), we get 2y + 4 = 0, so y = 2, whence x = ( 2) = 2. Therefore, the function has at most one maximum, minimum, or saddle, and any such point is at (2, 2). To determine what type of point, we have found, we use the second derivative test. X2 (2, 2) = 6 > 0, the point is a local minimum. Let r = {(x, y)|0 y ln(x), 1 x e}. We compute the double integral by integrating rst with respect to y and then with respect to x. (cid:90) (cid:90) 1 (cid:90) e (cid:90) e (cid:90) e (cid:90) e. 1 (cid:90) ln(x) (cid:90) ln(x) (cid:21)ln(x) (cid:20) y2.
