Magnetic mirror / electron motion and velocity

(F. F. ~ Chen 2.20) The magnetic field along the axis of a magnetic mirror is \(B_(z) = B_0(1+\alpha^2z^2)\), where \(\alpha\) is a constant. Suppose that at $z=0$ an electron has velocity \(v^2 = 3 v_{\parallel}^2 = \frac{3}{2}v_{\perp}^2\).

(a) Describe qualitatively the electron motion. (b) Determine the values of $z$ where the electron is reflected. (c) Write the equation of motion of the guiding center for the direction parallel to \(\vec{B}\) and show that there is a sinusoidal oscillation. Calcule the frequency of the motion as a function of \(v\).