Finding the absolute maximum

Question

You are in charge of manufacturing the snazzy new mobile tablets that everyone wants to own. The revenue function, in dollars, is given by

$R(s,t) = 8s+6t-s^2-2t^2+2st$ , s denotes "steel" model and t denotes "titanium" model, both in units of million (assume that you make positive but a finite number of products).

I have to determine the quantity of both products for maximum revenue.

My understanding:

So, I think the question is asking for the global maximum point. I found the critical point and it has only one, (11,7). Now, I think we need to assume that the lowest boundary for s and t is 0 and the upper boundary is also something (I don't know what to assume). And I'm stuck here.

Answer 1

I found the critical point and it has only one, (11,7).

This suggests trying to isolate relevant squares from the quadratic, and indeed a straightforward manipulation leads to the following form, which shows that there is one unique global maximum at $(11,7)$ without requiring any further assumptions: $\;R(s,t) = 65 - (t-7)^2 -(s - t - 4)^2\,$.