POEFS.A1.P3

eqn[w_] = LogicalExpand[D[(200 - q1 - q2) (q1 + q2) - w q1 - q2^2, {{q1, q2}}] == 0
                        && q1 >= 0 && q2 >= 0]
Manipulate[
 Column@{
   Piecewise[{{w, 2 q > w}, {2 q, 2 q <= w}}] == 200 - 2 q // Reduce[# && q >= 0, q] &,
   Which[w < 0,   Reduce[eqn[w][[{2, 3}]] /. q2 -> 0],
         w > 100, Reduce[eqn[w][[{1, 4}]] /. q1 -> 0],
         True,    Reduce[eqn[w]]]},
 {{w, 100}, -50, 200, Appearance -> "Labeled"}]