The discussion focuses on using the inverse function theorem to estimate changes in the roots of the cubic polynomial $p(\lambda) = \lambda^3 - 6\lambda^2 + 11\lambda - 6$ when the coefficients change. Specifically, it examines how the roots $0 < x_1 < x_2 < x_3$ vary when the coefficients $a = (a_2, a_1, a_0) = (-6, 11, -6)$ are altered by a small amount $\Delta a = 0.01a$. The method involves substituting $x_1$ into the polynomial and calculating the total derivative with respect to $x_1$ to apply the inverse function theorem effectively.
PREREQUISITESMathematicians, students studying calculus, and anyone interested in polynomial root analysis and perturbation methods will benefit from this discussion.
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?
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 $x_1$.