Don't quite understand question

[JG89]

Question: Let f: \mathbb{R}^3 \rightarrow \mathbb{R} be given by f(x,y,z) = sin(xyz) + e^{2x + y(z-1)}. Show that the level set \{ f = 1 \} can be solved as x = x(y,z) near (0,0,0) and compute \frac{\partial x}{\partial y} (0,0) and \frac{\partial x}{\partial z} (0,0,0).

I don't understand this question. I know what the level set is. It will be the set of points (x,y,z) such that f(x,y,z) = 1. I don't understand what they mean by, show that the level set can be solved as x = x(y,z) near (0,0,0). Any help?

 

[arildno]

Now, if f(x,y,z)=1, the implicit function theorem says that in some region D in the (y,z)-domain around a solution vector (x_{0},y_{0},z_{0}), we may define a function X(y,z), so that within D, the function F(y,z)=f(X(y,z),y,z)=1 for ALL (y,z) in D.

To take an example.

1. Let f(x,y,z)=x+y+z

Set f=1.

We now define: X(y,z)=1-y-z

Note that we then have:
F(y,z)=f(X(y,z),y,z)=(1-y-z)+y+z=1, irrespective of the values of (y,z).

This holds, for example, in some neighbourhood of the point (1,0,0).