Solution:

Let (v_{max}) be the maximum velocity. Time to accelerate: \(t_1 = \frac{v_{max}}{\alpha}). Time to decelerate: (t_2 = \frac{v_{max}}{\beta}). Total time (t = t_1 + t_2 = \frac{v_{max}}{alpha} + \frac{v_{max}}{beta} = v_{max}\left(\frac{1}{alpha} + \frac{1}{\beta}\right) = v_{max}\left(\frac{\beta + \alpha}{\alpha\beta}\right)). Solving for \(v_{max}): (v_{max} = \frac{\alpha\beta t}{\alpha + \beta}).