Suppose that the cost function of a monopolist changes from c1(y) to c2(y) with 0 < c1(y)

Question:

Suppose that the cost function of a monopolist changes from c1(y) to c2(y) with
0 < cʹ1(y) < cʹ2 (y) for every y > 0
Show that
c2(y1) - c2 (y2) > c1(y1) - c1(y2) (15)
where y*1 and y*2 are the profit maximizing output levels when costs are c1 and c2 respectively.