max(a_ and b_) := max(a,b);
max(sin(x_)):=1;
max(cos(x_)):=1;
max(-x_^2):=0;
max(a_ * b_) := If(a>0, a*max(b), If(a<0, a*min(b) ));
max(a_ + b_) := max(a)+max(b);
max(a_ - b_) := max(max(a)-max(b), max(a)+max(b));