Consider a plasma with a distribution function F(v) that has precisely two peaks, at v = v₁ and v = v₂ [with F(v₂) ≥ F(v₁)], and a minimum between them at v = v_min, and assume that the minimum is deep enough to satisfy the Penrose criterion for instability [Eq. (22.52)]. Show that there will be at least one unstable mode for every wave number k in the range k_min max, where

Show, further, that the marginally unstable mode at k = k_max has phase velocity ω/k = v_min, and the marginally unstable mode at k = k_min has ω/k = v₁.

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k² min p+∞⁰ F(v) - F(v₁) (v - v₁)² 1000⁰ tome -dv, k² = max e² fome p+∞ F(v) - F(min) (v - Umin)² 150⁰ dv. (22.52)