Let $g:\mathbb{R}_+\rightarrow\mathbb{R}$ is smooth, $g(1)=1$, and $g'(1)\ne \space \lambda$
(a)Prove that the set $S:=\{(x,y,z)\in \mathbb{R}_+^3 \vert \space x+g(x)+y+g(y)+z+g(z=6)\}$ is locally, in a neighborhood of the point $(1,1,1)$, the graph of a smooth function f in a neighborhood $(1,1)\subset \mathbb{R}_+^2$, provided $g'(1)\ne \space \lambda$ (Fill in correct number for $\lambda$)
(b)Calculate the first and second partial derivatives of f at (1,1), in terms of corresponding quantities for g.
(c)Does $S$ lie inside or outside the sphere $x^2+y^2+z^2=1$, at least in the neighborhood of the point $(1,1,1)$: To answer the question, fins out whether $x^2+y^2+f(x,y)^2$ has a local min or max at $(1,1)$. However, specifically answer the question for the cases $g(t)=t^3$ and $g(t)=t^{3/2}$
Here is what I have so far:
for a: I see that $S = \{ (x,y,z) \in \Bbb R^3 \mid F(x,y,z) = 6 \}$, where $F\colon \Bbb R^2 \times \Bbb R_{\geq 0} \to \Bbb R$ is given by $F(x,y,z) = x+g(x)+y+g(y)+z+g(z)$. We have that $F(1,1,1) = 6$. Now, I need to compute $D_z F(1,1,1)$. I feel as though the Implicit Function Theorem applies but i'm sure how
for b: Then we have $F(x,y,z(x,y)) = 6$ for all $x$ and $y$ near $(1,1)$, which is sufficient to allow us to apply partial derivatives.
for c to see whether $x²+y²+z(x,y)²$ has a maximum or minimum at $(1,1)$, I believe the Hessian to be the correct way to go about this but it doesn't seem to work out for me