If the average (arithmetic mean) of seven distinct positive integers

Sum of the 7 integers = (count)(average) = 7*14 = 98

The 7 distinct integers must sum to 98.
For the greatest integer to be AS SMALL AS POSSIBLE, the other 6 integers must be AS LARGE AS POSSIBLE.
Implication:
The 7 distinct integers must be CONSECUTIVE, so that the 6 smallest integers are as large as possible without exceeding the value of the largest integer.
To illustrate:
101, 102, 103, 104, 105, 106, 107
In this set of 7 distinct integers, the six smallest integers (101 through 106) are as large as possible without exceeding the value of the largest integer (107).

Consecutive integers constitute an EVENLY SPACED SET.
For any evenly spaced set:
average = median

In the posted problem, the average -- and thus the median -- of the 7 consecutive integers is 14:
__ __ __ 14 __ __ __
Since the median of the seven integers is 14, the three largest integers must be 15, 16, and 17.
Thus, the largest integer = 17.