# Intrinsic critical points of a function .

1. Nov 29, 2004

### Saint Medici

Intrinsic critical points of a function.....

I have a problem that's giving me a bit of trouble. The solution is not vital to my existence, but it's been eluding me long enough that it's grown to be a pest. I've figured out the easy part by hand, and I have the less obvious solutions by way of TI-89, but I'd like to know a way to get there without computer assistance. Anyway:

Find all the intrinsic critical points of $$f(x)=x_1^3+2x_1x_2x_3-x_3^2$$ on the unit sphere.

Of the ten, I've found six, namely, all the unit vectors along the coordinate axes. We've been using the Lagrange formulation quite a bit, so I assume that's what we're to use here. Anyway, any insight would be appreciated.