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

#### ianchenmu

##### Member
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?

#### Klaas van Aarsen

##### MHB Seeker
Staff member
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 $x_1$.

#### ianchenmu

##### Member
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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