Another implicit function problem

[Srumix]

I'm having some trouble grasping the implicit function theroem in some cases. Here's one of them.

Homework Statement

Show that there exist a C² function y(x) in some neighbourhood of 0 such that y(0) = 0 and

y(x)³ + 3y(x) = x

Find y'(0) and y''(0)

Homework Equations

The implicit function theorem: If there exist a implicit y(x) for function if D2f(a,b) does not equal zero at the point (0,0)

The Attempt at a Solution

I tried differentiating the given function but i then get 3y(x)²*y'(x) + 3*y'(x) = 0.

But given the condition y(0) = 0 this must imply that y'(0) = 0 which is a contradiction?

The problem is given in a way such that there should be such a function, but I'm obviously doing something wrong here.

 

[hunt_mat]

There are two ways that you can go about this, there is the easy way which as you say, use the implicit function theorem (you have to use partial derivatives, you need to calculate the whole matrix of derivatives btw.) and for that you need to calculate partial derivatives, not the total one, the derivative you got wrong, it should be 3y^{2}(x)y'(x)+3y'(x)=1, if you're differentiating w.r.t x.

The other way is to solve the equation for y(x) by using the standard solution for a cubic and verify that the function is indeed C^2 in the neighbourhood of 0.

 

[Srumix]

Oh yes. Kind of stupid of me to mess upp the derivative.

When you say the matrix of derivatives, do you mean the "jacobian"? Should I calculate the Jacobian first for the first partial derivative and then the jacobian for the second derivative to check that the determinant of the jacobian isn't zero?

Thanks!

 

[hunt_mat]

Check out the conditions for the implicit function theorem. You will need to check out the Jacobian.

 

[Srumix]

Check out the conditions for the implicit function theorem. You will need to check out the Jacobian.

Yeah I think I understood that. However I'm having trouble of how I should do to check that it is actually C². To check that it is C², should I insert D2f(x,y) in the second row of the jacobian and keep the first row the same as I did when checking for C¹?

 

[hunt_mat]

Check out the Hessian matrix then.

 

[Srumix]

Check out the Hessian matrix then.

I have not much experience with the Hessian. Is it required (in analogue to the Jacobian) that it's determinant isn't equal to zero for the implicit function to be defined as C²? Or is there any other general method to determine wether there exist a C² function.

Thanks for your help!