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# Maximum and Minimum

Let f be a function of one variable defined for all x in some domain D. A global maximum of f is a point in D such that for all x in D.

For a constant , the neighborhood of a point is the set of all points x such that . A point is a local maximum of f if there exists such that for all x in where f(x) is defined.

Figure 2.1: local maxima and minima

Similarly one can define local and global minima. In Figure 2.1, the function f is defined for x in domain [ a , e ]. Points b and d are local maxima and b is a global maximum, whereas a, c, and e are local minima and e is a global minimum.

Finding extrema

Extrema, whether they are local or global, can occur in three places:

1. at the boundary of the domain,

2. at a point without a derivative, or

3. at a point with .

The last case is particularly important. So we discuss it further. Let f be differentiable in a neighborhood of . If is a local extremum of f, then . Conversely, if , three possibilities may arise: is a local maximum, is a local minimum, or neither!!! To decide between these three possibilities, one may use the second derivative. Let f be twice differentiable in a neighborhood of .

• If and then is a local minimum.
• If and then is a local maximum.
• If and then may or may not be a local extremum.

Figure 2.2 illustrates these three possibilities.

Figure 2.2: , and one of the possibilities with

The revenue at price x is

We compute the derivative of g and set it to 0.

Since f(x);SPMgt;0 for all x, setting g'(x)=0 implies 1-0.2x=0. So x=5. This is the only possible local optimum. To determine whether it is a maximum or a minimum, we compute the second derivative.

Putting in x=5 shows g''(x);SPMlt;0, so this is a maximum: the oil cartel maximizes its revenue by pricing gasoline at \$5 per gallon.

We must first calculate the holding cost (cost of stockpiling). If each order period begins with Q cases and ends with zero cases, and if usage is more or less constant, then the average stock level is Q/2. This means that the average holding cost is hQ/2.

Since each order of Q cases lasts Q/d weeks, the average ordering cost per week is Thus the average total cost per week is

To find Q that minimizes cost, we set to zero the derivative with respect to Q.

This implies that the optimal order quantity is

This is the classical economic order quantity (EOQ) model for inventory management. (You will learn when the EOQ model is appropriate and when to use other inventory models in 45-765).

Next: Binary Search Up: Unconstrained Optimization: Functions of Previous: Derivatives

Michael A. Trick
Mon Aug 24 16:30:59 EDT 1998