# Multivariable Minimization Question

1. Apr 19, 2006

### SigurRos

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.

2. Apr 19, 2006

### SigurRos

Good news:
I got the solution.