My goal is to get an analytical solution of x ( $\lambda$ in the problem), when $\epsilon$ (f[x] in the problem) is maximum (use inverse form).
I found that if I can use trigonometric identity to change
Cos[2x], then I can achieve my goal. Here is the code that I wrote, but I found it is not very good, because for the third line I copy and paste from the evaluation cell.
So, can anyone think of an easier way to change $\epsilon$ in terms of 2$\lambda$, or can provide another way to achieve the goal?
f[x_] = (R ω^2 Sin[x] Cos[x])/(g - ω^2 (Cos[x])^2 R); f'[x] // FullSimplify; Solve[2 R ω^2 (R ω^2 + (-2 g + R ω^2) Cos[2 x]) == 0, Cos[2 x]][] // FullSimplify;