One recommendation I have is to differentiate the function with respect to the parameters (i.e. f(b,d,p) > 0) and use that to show that the derivatives of the function f have the right properties to follow (i.e. with respect to turning points and derivatives).
If you can for example show that you have monotonicity or a similar property over the range of the variables (like p) then you can show that the function is not negative.
For multi-variable scenarios you will need to look at the Hessian in the worst case and the single partial derivatives in the easiest case.
Since you have restrictions on your range, you can evaluate the appropriate derivatives at the extremities to see what the behavior is across the range.