Finding the associated parameter of the maximum

Question

I am trying to find the maximum value in a list. I can find the maximum value, but how can I find the value of the associated parameter at the maximum? (I don't want the index of the element in the list of the parameter since the actual list (of t in the example) is so long, so I want to get t=4 for the following example instead of its index:3)

 v = Table[
  TR1 + TR2 /. Flatten[{Solve[TR1 - t == 0, TR1], TR2 -> t + 1}], {t, 
   2, 4, 1}]

tstar = Max[v]

Answer 1

t = Range[2, 4, 1]
v = (TR1 + TR2 /. 
     Flatten[{Solve[TR1 - # == 0, TR1], TR2 -> # + 1}] &) /@ t
t[[Ordering[v, -1]]]

Answer 2

For you example, the following will work.

 Maximize[{1 + 2 t, 1 <= t <= 4 && t ∈ Integers}, t][[2, 1, 2]]

4

It will work for functions that are not monotonic as well.

Maximize[{Sin[π t/2], 0 <= t <= 4 && t ∈ Integers}, t][[2, 1, 2]]

1

In the above two examples the global maximum falls on an integer value, but this does not have be the case. In the following example, the global maximum is at π/2, but the integer maximum is at 2.

Maximize[{Sin[ t], 0 <= t <= 4 && t ∈ Integers}, t][[2, 1, 2]]

2