Chapter 7.6, Problem 3P

Problems 1 through 3 ask you to fill in some of the details of the analysis of the Lorenz equations in this section:

(a) By solving Eq. (9) numerically, show that the real part of the complex roots changes sign when r ≅ 24.737.

(b) Show that a cubic polynomial $x^3+A{x}^2+Bx+C$ has one real zero and two pure imaginary zeros only if $AB=C$.

(c) By applying the result of part (b) to Eq. (9), show that the real part of the complex roots changes sign when $r=\frac{470}{19}$.

$3{\lambda }^3+41{\lambda }^2+8\left(r+10\right)\lambda +160\left(r-1\right)=0,\left(9\right)$