ABSTRACT
We are all quite familiar, from courses in mathematics, with the determination of the maximum or minimum of a function by the use of calculus. If the function is continuous and differentiable, its derivative becomes zero at the extremum. For a function y(x), this condition is written as
dy dx = 0
(8.1)
where x is the independent variable. The basis for this property may be explained in terms of the extrema shown in Figure 8.1. As the maximum at point A is approached, the value of the function y(x) increases and just beyond this point, it decreases, resulting in zero gradient at A. Similarly, the value of the function decreases up to the minimum at point B and increases beyond B, giving a zero slope at B. In order to determine whether the point is a maximum or a minimum, the second derivative is calculated. Since the slope goes from positive to negative, through zero, at the maximum, the second derivative is negative. Similarly, the slope increases at a minimum and, thus, the second derivative is positive. These conditions may be written as (Keisler, 1986)
For a maximum:
d y dx
2 0<
(8.2)
For a minimum:
d y dx
2 0>
(8.3)
These conditions apply for nonlinear functions y(x) and, therefore, calculus methods are useful for thermal systems, which are generally governed by nonlinear expressions. However, both the function and its derivative must be continuous for the preceding analysis to apply.
