I am having difficulty getting Maple to find the optimal solution over a specified range. That is, I want to limit the possible range of my maximizers. For my problem I am maximizing the expression "payoff" with respect to p and s and want to tell Maple to search for all solutions where s<pa.

I have the following expression for variable x

x:=-1/4*(s^2-pa^2+4*pa-4*s-4*p)/(s-pa);

where x is an input to the function payoff that I wish to maximize:

payoff:=p*x + s*subs(Q=x,y) + m*(1-subs(Q=x,y)));

and

y:=int((d-s/2),d=s/2..Q+s/2)+int(Q,d=Q+s/2..1));

Now, defining the FOC of payoff as