Implicit function theorem

[mathematicianinprogress]

Hello, I've been thinking about this problem but I have no idea how it is:

If x₁, x₂ and x₃ are the roots of the polynomial p(x)=x³+y₁x²+y₂x+y₃, there is this relationship with the coefficients:

y₁=-(x₁+x₂+x₃)
y₂=x₁x₂+x₁x₃+x₂x₃, (Cardano-Vieta's equations)
y₃=-x₁x₂x₃.

Prove that there is an entourage of real roots (a, b, c), different two by two, where a C¹ function, which roots are expressed in terms of the coefficients, is defined. Calculate one of the partial derivatives of one of the components of that function.

Thanks in advance!

 

[Steven G]

This being a math help forum you really should not expect any to solve this for you. On this forum the helpers expect you to be involved in solving your problem while receiving hints from the forum helpers. This requires you to show what you have done so far so the helpers can get you back on track. All this is in the forum guidelines (did you read them??) and if you had followed them you would have received help by now.

 

[Dr.Peterson]

Prove that there is an entourage of real roots (a, b, c), different two by two, where a C¹ function, which roots are expressed in terms of the coefficients, is defined. Calculate one of the partial derivatives of one of the components of that function.

The first step is to figure out what this means. What does "entourage" mean to you? How is "different two by two" different from "distinct"? What does "where ... is defined" mean?

I wonder if you are translating this from another language, and we need to see the original, or a better translation.

 

[LCKurtz]

I think the OP is wanting to show the roots vary continuously in terms of the coefficients by showing the existence of a local [MATH]C^1[/MATH] multivariable function expressing the roots in terms of the coefficients, using the multivariable implicit function theorem.