# LaGrange Multiplier

1. Mar 15, 2005

### jenc305

Find the points at which the function f(x,y,z)=x^8+y^8+z^8 achieves its minimum on the surface x^4+y^4+z^4=4.

I know
8x^7=(lamda)4x^3
8y^7=(lamda)4y^3
8z^7=(lamda)4z^3
x^4+y^4+z^4=4

Case1: x not equal to 0, y not equal to 0, and z not equal to 0
I get 3(4th root of 4/3 to the eigth)???

I'm I doing this right?

2. Mar 15, 2005

### HallsofIvy

Staff Emeritus
Are you sure this doesn't belong in the homework section?

Since grad f always points in the direction of fastest increase of f, to get to a minimum, you should go in the exact opposite direction. Of course, if you are required to stay on the surface x4+ y4+z4= 4, you can't do that. What you can do is go opposite to the component tangent to the surface. That works until you get to a point where grad f is perpendicular to the surface and has no component tangent- that's when you are at the minimum point on that surface.
If you let g(x,y,z)= x4+ y4+ z4 then the surface is a level surface of g- and grad g is perpendicular to the surface: we must have grad f parallel to grad g which means grad f= &lamda; grad g for some number &lamda;. That's the idea of the LaGrange multiplier.