The difficulty with this reasoning is the assumption that the minimizing function u must have two derivatives.

Suppose one has a function u which describes the temperature at a given location (x, y, z).

The heat equation is used to determine the change in the function u over time.

The function u above represents temperature of a body.

The function u(t) is simply determined by the Pythagorean theorem:

In the next group of examples, the unknown function u depends on two variables x and t or x and y.

The function u(w) mentioned earlier can therefore be considered to consist of differing sets of valued time.

The function U(x) is called the potential energy associated with the applied force.

This function U is an indispensable tool used in the analysis of many physical systems.

Conversely, any agent acting to maximize the expectation of a function u will obey axioms 1-4.