  In the present paper, the maximum principle for finite horizon state
constrained problems from the book by R. Vinter [\textit{Optimal Control},
Birkh\"auser, Boston, 2000; Theorem~9.3.1] is analyzed via parametric examples.
The latter has origin in a recent paper by V.~Basco, P.~Cannarsa, and
H.~Frankowska, and resembles the optimal growth problem in mathematical
economics. The solution existence of these parametric examples is established
by invoking Filippov's existence theorem for Mayer problems. Since the maximum
principle is only a necessary condition for local optimal processes, a large
amount of additional investigations is needed to obtain a comprehensive
synthesis of finitely many processes suspected for being local minimizers. Our
analysis not only helps to understand the principle in depth, but also serves
as a sample of applying it to meaningful prototypes of economic optimal growth
models. Problems with unilateral state constraints are studied in Part 1 of the
paper. Problems with bilateral state constraints will be addressed in Part 2.
