Old/Practice Second 4 O’clock Exam SolutionsMath 231.Find the direction of greatest increase of the functionf(x, y) =x2y+ 2y2at thepoint(1,2).
2.Find the equation of the tangent plane to the surfacex2-2xy+ 4y2+z2= 14at thepoint(1,2,-1).
3.Find the absolute maximum and absolute minimum values of the functionf(x, y) =3x2+y2-2yon the regionRdescribed byx2+y2≤4.First, we look for interior critical points. Since∇f= 6x~ı+ (2y-2)~,certainly∇f= 0 only when (x, y) = (0,1), and this point is in the regionR.Next, we consider the boundary,x2+y2= 4.You can either parametrize the bound-ary byx= 2 cos(t) andy= 2 sin(t) and find the critical points off(2 cos(t),2 sin(t)) =12 cos2(t) + 4 sin2(t)-4 sin(t) fort∈[0,2π], or you can use Lagrange multipliers. Using La-grange multipliers, we realize the boundary as thek= 4 level curve ofg(x, y) =x2+y2. Thenthe critical points on the boundary come from solutions to the system6x= 2λx2y-2 = 2λyx2+y2= 4