Design a Discrete-Time System
Since you have defined your signal $x[n]$ to be a set of non-negative numbers with a single global maxima, the algorithm to find the maximum is quite simple.

compute the first (causal , backward) difference 
$$y[n] = x[n] - x[n-1]$$

Then look for the sign change of $y[n]$; i.e., at the extreme point you will have the sign of $y[n]$ changing from positive to negative,  assuming you had a single global maxima.

$$z[n] = \text{sign_change_detector} \{ y[n] \} $$

Don't forget to check the end points as well. As they are also the candidates to be maximum in addition to those critical point(s).