I have a question kind of related to comparative statics in economics while I was doing some economic research myself. I would really appreciate if you could provide some insights.

1. f(x1,x2) is defined on a compact set, assume the extrema is not on the boundary, therefore in the interior, denoted by (x1*, x2*) 2. Suppose there is an exogenous variable t in the function f(x1,x2), I wanted to see how x1* changes as t changes. What I am thinking is that since I have proved the existence of the extrema, I can take it into the FOC, and gives me the following equation. x1f(x1,x2))=0 Then I can use the similar essence of implicit differentiation, where I take derivative with respect to t on both sides. The reason why I use the implicit differentiation is because I can't explicitly isolate x1* in the above equation. 3. I have also calculated Hessian matrix that is strictly concave, meaning it's a global maximum. 4. But when I calculate the Hessian matrix, I only consider the two variables x1, x2, shouldn't I also consider the exogenous variable t, where I need to consider a 3 by 3 Hessian matrix? And if I only consider the Hessian matrix with x1,x2, wouldn't it make t as a constant? If t is a constant, wouldn't it make no sense to take derivative with respect to it? Also, is step 2 valid step to find how x1* changes as t changes?