I'm reading the solution of an optimization problem where i have to optimize the volume of a cylinder according to a specific cost.
The cost is $c_0 = 2\pi r^2c1 + 2\pi rhc_2 = 2\pi r^2c_1 + \pi rhc_2 + \pi rhc_2 = 3(\frac{2\pi r^2c_1 + \pi rhc_2 + \pi rhc_2}{3}) \ge 3(2\pi \pi^2 r^4 h^2 c_1c_2^2)^{1/3}=3(2\pi V^2c_1c_2^2)^{1/3}=3(2\pi)^{1/3}(c_1c_2^2)^{1/3}V^{2/3}$
But i don't understand the part after the inequality, i know that the AG inequality is used but i don't know how, why does the denominator disappear and how do we get $2\pi \pi^2 r^4 h^2 c_1c_2$ ?