Implicit function theorem part 2

  • #1
Lambda96
189
65
Homework Statement
see Post
Relevant Equations
Implicit function theorem
Hi,

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

Bildschirmfoto 2024-07-06 um 16.56.27.png


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)##
 
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  • #2
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.
 

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