MHB Use the inverse function theorem to estimate the change in the roots

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Let $p(\lambda )=\lambda^3+a_2\lambda^2+a_1\lambda+a_0=(\lambda-x_1)(\lambda-x_2)(\lambda-x_3)$ be a cubic polynomial in 1 variable $\lambda$. Use the inverse function theorem to estimate the change in the roots $0<x_1<x_2<x_3$ if $a=(a_2,a_1,a_0)=(-6,11,-6)$ and $a$ changes by $\Delta a=0.01a$. How can I use the inverse function theorem to estimate?
 
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ianchenmu said:
Let $p(\lambda )=\lambda^3+a_2\lambda^2+a_1\lambda+a_0=(\lambda-x_1)(\lambda-x_2)(\lambda-x_3)$ be a cubic polynomial in 1 variable $\lambda$. Use the inverse function theorem to estimate the change in the roots $0<x_1<x_2<x_3$ if $a=(a_2,a_1,a_0)=(-6,11,-6)$ and $a$ changes by $\Delta a=0.01a$. How can I use the inverse function theorem to estimate?

You could start by filling in $x_1$ in $p(\lambda )=\lambda^3+a_2\lambda^2+a_1\lambda+a_0=0$ and taking the (total) derivative with respect to $x_1$.
 
I like Serena said:
You could start by filling in $x_1$ in $p(\lambda )=\lambda^3+a_2\lambda^2+a_1\lambda+a_0=0$ and taking the (total) derivative with respect to $x_1$.

What $\Delta a$ means? Can you give me a more complete answer? Thank you.

Who can provide me a complete answer?
 
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A sphere as topological manifold can be defined by gluing together the boundary of two disk. Basically one starts assigning each disk the subspace topology from ##\mathbb R^2## and then taking the quotient topology obtained by gluing their boundaries. Starting from the above definition of 2-sphere as topological manifold, shows that it is homeomorphic to the "embedded" sphere understood as subset of ##\mathbb R^3## in the subspace topology.

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