I got this question as a take home exam question, and I can't figure it out for the life of me: The temperature T(x,y,z) throughout a region in space is given by: T(x,y,z) = 3*x^2*y*2+z^2 An insect is confined to move on the surface S : x^2 + y^2 = z. The insect is at the point P(1,1,2) on S and wishes to move in the direction in which T decreases the fastest. However, the insect can only move in directions tangential to S. In which direction should the insect move from P(1,1,2)? Here's what I've got so far: The Temperature will decrease the fastest in the direction opposite to the Gradient of T(x,y,z), which is: -<6*x*y^2 x, 6*x^2*y y, 2*z z> , which at (1,1,2) is equal to <-6 x, -6 y, -4 z>. The equation for the tangent plane at (1,1,2) is: PLANE: 2*(x-1)+2*(y-1)-(z-2) = 0 My professor says that Lagrange Multipliers may be used, but I'm not quite sure how. I'm not sure if the constraints are the plane, the surface S, or both. I tried computing using both the plane and S as constraints, but it didnt work because they only touch at (1,1,2). Any suggestions? This is due in exactly 12 hours from right now, any help will be greatly appreciated.