Implicit function theorem part 2

[Lambda96]

Homework Statement

see Post

Relevant Equations

Implicit function theorem

Hi,

I'm not sure if I've solved the problem correctly

In order for the Implicit function theorem to be applied, the following two properties must hold $F(x_0,z_0)=0$ and $\frac{\partial F(x_0,z_0)}{\partial z} \neq 0$. $(x_0,z_0)=(1,2)$ is a zero and $\frac{\partial F(x_0,z_0)}{\partial z} =-x^2=-1$ so both properties are fulfilled


According to the Implicit function theorem, there now exists a function $g$ for which the following relation $x=g(z_1)$ is valid in the neighborhood $(z_0 - \epsilon, z_0 + \epsilon)$.


Then the following applies $F(x,z_1)=F(g(z_1),z_1)$

 

[docnet]

You're doing good! keep going. You could end up with a value of some $\epsilon>0$ in the end, because that's the the problem is asking for. Then, you could show that the statement in the problem is true using the value of $\epsilon$ you found and the implicit function theorem.