Finding C^1 functions satisfying four equations near (2,1,-1,-2)

[mistahkurtz]

Homework Statement

Just a clarification: the two last equations must hold in an open neighborhood of the point (2, 1, -1, -2), not just at that point.

Homework Equations

The Attempt at a Solution

I have to do an existence proof. The shortest way of accomplishing this would just be to just construct two C^1 functions that satisfy the requirements:

(1) f1(2,1,-1,-2)=4
(2) f2(2, 1, -1, -2)=3
(3) f1^2 + f2^2 + x4^2 = 29
(4) f1^2 / x1^2 + f1^2 / x2^2 + x4^2 / x3^2 = 17

But I've tried unsuccessfully to solve for f1 and f2 from the above equations, and at just guessing and checking.

I guess the strategy I have to take is to simply assume f1 and f2 are functions satisfying (1)-(4), and then show that they can indeed be continuously differentiable. However, I'm not exactly sure if this is the right way to proceed.

 

[Eynstone]

The equations 3 & 4 is a system of linear equations in f1^2 , f2^2.