SOLUTION FOR IRODOV PROBLEM 4.34 -- two masses interconnected by a spring -- one body performs free vertical harmonic oscillations -- maximim and minimum force

I have slightly altered the problem so that it is general in its composition; I must admit that, in spite of looking up to the answer, I had nearly spent half-a-day in understanding the solution completely. Now here is the problem:
A body A of mass m₁ and a body B of mass m₂ are interconnected by a spring as shown in Fig. 2. The body A performs free vertical harmonic oscillations with amplitude A and angular frequency ω. Neglecting the mass of the spring, find the maximum and minimum values of force that this system exerts on the bearing surface.  
Solution: