why was the extrapolated final temperature (and not the maximum or minimum recorded temperature) used to determine the heat of solution?

Extrapolated final temperature of the solution is stable and realistic.

Explanation:

When two or more chemicals interact in the solution, the checmial reaction is generating heat, or even absorbing it from the environment. This is called heat of solution and this minimum or maximum recorded temperature is not stable. Initial mixture is not well-mixed and the certain period of time is needed to pass, so the solution may reach some sort of stability and homogeneity. However, after long period of time, the temperature will be constant. Extrapolated final temperature represents the final temperature of the solution and equilibrium temperature, all of which depends on solution behavior and characteristics.