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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 tex2html_wrap_inline488 in D such that tex2html_wrap_inline492 for all x in D.

For a constant tex2html_wrap_inline498 , the neighborhood tex2html_wrap_inline500 of a point tex2html_wrap_inline488 is the set of all points x such that tex2html_wrap_inline506 . A point tex2html_wrap_inline488 is a local maximum of f if there exists tex2html_wrap_inline498 such that tex2html_wrap_inline492 for all x in tex2html_wrap_inline500 where f(x) is defined.

Figure 1.1: local maxima and minima

Similarly one can define local and global minima. In Figure 1.1, 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 tex2html_wrap_inline488 with tex2html_wrap_inline544 .

The last case is particularly important. So we discuss it further. Let f be differentiable in a neighborhood of tex2html_wrap_inline488 . If tex2html_wrap_inline488 is a local extremum of f, then tex2html_wrap_inline544 . Conversely, if tex2html_wrap_inline544 , three possibilities may arise: tex2html_wrap_inline488 is a local maximum, tex2html_wrap_inline488 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 tex2html_wrap_inline488 .

Figure 1.2 illustrates these three possibilities.

Figure 1.2: tex2html_wrap_inline376 , tex2html_wrap_inline378 and one of the possibilities with tex2html_wrap_inline380


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 tex2html_wrap_inline634 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 up previous contents
Next: Binary Search Up: Unconstrained Optimization: Functions of Previous: Derivatives

Michael A. Trick
Mon Aug 24 13:43:30 EDT 1998