only answer d) and e) thanks

Consider the Lorenz system z' =0(y - x), y = px - y - xz, 1 = ry - B2 with o >0, p > 0 and B > . (a) Show that this system is invariant under the transformation (2, y, z) +(-1, -y, z, t). (b) Show that z-axis is invariant under the flow of this system and that it consists of three trajectories. (c) Show that this system has the unique equilibrium point at the origin for p 1. (d) Find the eigenvalues of linearized system at the origin of the Lorenz equations and then show that the origin is asymptotically stable for p 1. (e) Use the Lyapunov function V(x, y, z) = px? + oy2 + oz2 to show that the origin is globally asymptotically stable for p