Maxima and minima pop up all over the place in our daily lives. They can be found anywhere we are interested in the highest and/or lowest value of a given system; if you look hard enough, you can probably find them just about anywhere! Here are just a few examples of where you might encounter maxima and minima:
*A meteorologist creates a model that predicts temperature variance with respect to time. The absolute maximum and minimum of this function over any 24-hour period are the forecasted high and low temperatures, as later reported on The Weather Channel or the evening news.
* The director of a theme park works with a model of total revenue as a function of admission price. The location of the absolute maximum of this function represents the ideal admission price (i.e., the one that will generate the most revenue).
* An actuary works with functions that represent the probability of various negative events occuring. The local minima of these functions correspond to lucrative markets for his/her insurance company - low-risk, high-reward ventures.
* A NASA engineer working on the next generation space shuttle studies a function that computes the pressure acting on the shuttle at a given altitude. The absolute maximum of this function represents the pressure that the shuttle must be designed to sustain.
* A conscientious consumer meticulously collects data on his/her cell phone usage over a certain interval (say, a month) and develops a function to represent cell phone usage over time. The local maxima and minima of this function give the consumer a better idea of his/her usage patterns, empowering him/her to choose the most appropriate cell phone plan.
Local Maxima and Minima
A function y = f(x) has a local maximum at a point when the y-value at that point is greater than at any other point in the immediate neighbourhood. There may be larger values somewhere else but standing at the local maximum the ground falls away in both directions.
A local minimum is a point where the y-value is less than at any other point in the immediate neighbourhood. The plural of "maximum" and "minimum" are "maxima" and "minima".
If there's a derivative at a local maximum or minimum it clearly must be zero because a positive or negative slope would mean that the curve is higher on one side or the other. But there's also the possibility that a function has no derivative at the local maximum or minimum.
The following graph clearly has a local minimum at the vertex but at that point there's no uniquely defined slope.
There's no tangent at this minimum point and so it doesn't make sense to talk of the slope there. but we won't encounter this sort of situation very often so to find the local maxima and minima we simply put dy/dx = 0 and solve for x.
Stationary points are where the slope is zero or, in other words, dy/dx = 0. They include local maxima and minima, but here's another possibility. Here you see a point where the derivative is zero, but it's neither a local maximum nor a local minimum. We call it a stationary point of inflection. So there are three types of stationary point: local maxima, local minima and stationary points of inflection. Usually when asked to find the stationary points you'll be asked to classify them. This means to determine what type of stationary point they are.
More Terminology
Definition: A critical point of f(x) is a point where f0(x) = 0 or f0(x) is undefined.
Definition: A function f(x) has an absolute maximum at x = c if f(x) is the largest value of f(x). The f(c) is the maximum value of f(x). Similarly, f(x) has an absolute minimum at x = c if f(c) is the smallest value of f(x). Then f(c) is the minimum value of f(x).
The function f(x) has a local maximum at x = c if f(c) _ f(x) for all values of x near c and in the domain of f(x). Similiarly, f(x) has a local minimum at x = c if f(c) _ f(x) for all values of x near c and in the domain of f(x).
Finding Local Maxima and Minima
There are three possible places for local maxima and minima to occur:
1. At a value x = c where f0(x) = 0,
2. At a value x = c where f is not differentiable, or
3. At a value x = c that is an endpoint of the domain of f.
Notice that these are critical points of the function. Note that a critical point is not always a local max or min.
Once we have a point x = c where local maxima and minima might occur, how do we dermine whether the function has a local maxima, local minima, or neither at this point?
We have two tests that help us out :
First Derivative Test: Suppose y = f(x) has a critical point at x = c and that f is
continuous at x = c.
1. If f0(x) > 0 for values of x less than c and f0(x) < 0 for values of x greater than c, then x = c is a local maximum.
2. If f0(x) < 0 for values of x less than c and f0(x) > 0 for values of x greater than c, then x = c is a local minimum.
3. Otherwise, x = c is neither a local maximum nor a local minimum.
Second Derivative Test: Suppose y = f(x) has a critical point at x = c and that f00(c)
is defined.
1. If f00(c) < 0, then f has a local maximum at x = c.
2. If f00(c) > 0, then f has a local minimum at x = c.
3. If f00(c) = 0, we can't say anything with this test.
The Second Derivative Test
The simplest test to understand, and often the easiest to use, is the so-called "Second Derivative Test". There are three possibilities for the value of y/d at a stationary point. It can be positive, negative or zero. And there are three types of stationary point: maximum, minimum and stationary point of inflection. It would be tempting to suppose that the three possibilities for the value of y/d correspond to three types of stationary point, but unfortunately it's not quite that simple.
If y/d < 0 this means that the derivative of the derivative is negative, or in other words, the derivative is decreasing. Since it's zero at the stationary point this means that the slope must be positive to the left of the point and negative to the right. Clearly the stationary point must be a local maximum.
If y/d> 0 this means that the derivative of the derivative is positive, or in other words, the derivative is increasing. Since it's zero at the stationary point this means that the slope must be negative to the left of the point and positive to the right. Clearly the stationary point must be a local minimum.
SECOND DERIVATIVE TEST
For a Stationary Point (where dy/dx = 0):
Sign ofy/d
Nature of Stationary Point
+
Local MINIMUM
-
Local MAXIMUM
0
TEST FAILS !!
The First Derivative Test
Suppose we have a stationary point to y = f(x) at x = a. We can determine the nature of this stationary point by examining the sign of dy/dx immediately to the left of "a" and immediately to the right.
If dy/dx is negative for x < a and dy/dx is positive for x > a then the graph comes down, levels out momentarily, and then climbs again. This is a clear case of a local minimum. But before going on to the other cases we'd better clarify what we mean. We're referring only to points immediately to the left of x = a and immediately to the right of x = a. It doesn't matter what the value of dy/dx is further away. So long as, for some b < a we have dy/dx < 0 for b < x < a (that is, for x between b and a) and, for some c > a we have dy/dx > 0 for a < x < c, then there's a local minimum at x = a. Now all that is a bit of a mouthful. So instead we write:
If dy/dx < 0 for x = a-and dy/dx > 0 for x = a+ then there's a local minimum at x = a.
The phrase "x = a-" is shorthand for "x = a minus-a-little-bit" or, more precisely, what we said above: "for some b < a we have dy/dx < 0 for b < x < a".
Using this useful piece of shorthand we can identify four possibilities. By working out which one applies to a given example we can determine the nature of the stationary point.
FIRST DERIVATIVE TEST
For a Stationary Point (where dy/dx = 0):
Sign of dy/dx for x = a-
Sign of dy/dx for x = a+
Nature of Stationary Point
-
+
Local MINIMUM
+
-
Local MAXIMUM
-
-
Stationary Point of Inflection
+
+
Now rather than having to remember the above table (it's so easy to get the cases mixed up) you should simply draw the appropriate sketch, as indicated. Of course the actual curve won't be a series of straight lines as shown here. These are simply crude approximations to the curve to help us decide on the nature of the stationary point.
Of dy/dx immediately to the left of the point and immediately to the right. Where dy/dx is negative, draw a short downhill slope. Where dy/dx is positive draw a short uphill slope. The resulting picture will then indicate the nature of the stationary point. You may have wondered what happens if dy/dx is zero for all points immediately to the left, or immediately to the right of a stationary point. In such cases we have a whole region of stationary points. For example the horizontal straight line x = 2 consists of nothing else but stationary points. And these are neither local maxima nor minima. But nor are they stationary points of inflection. These types of stationary points are less common and not very useful, so we don't give them special names.
Which Should We Use: The First or Second Derivative Test?
The Second Derivative Test is more straightforward than the First and should always be the one you think of first. However it suffers from a few serious disadvantages.
For a start, the Second Derivative Test requires you to differentiate twice. This isn't much of a problem with the simple functions we've used so far but there are more complicated functions where it's difficult enough to find the first derivative, and a nightmare to differentiate a second time. If you find yourself having to work hard to differentiate the derivative again you should consider using the First Derivative Test instead.
The other serious deficiency of the Second Derivative Test is that it sometimes shrugs its shoulders and says, "I don't know!" This is when the second derivative is zero at the stationary point. Perhaps it's a stationary point of inflection. On the other hand it might be a local maximum or minimum. Stumped! In such cases, whether you like it or not, you have to fall back on the First Derivative Test.
The Second Derivative Test is often simple and effective BUT
it's sometimes inconclusive
it can involve much unnecessary calculation
and it disguises what's going on.
Global Maximum and Minimum
Local maxima and minima are all very interesting. But more usually we want to find a global maximum or minimum. This is the overall largest or smallest value of y over an entire range of values.
Suppose a manufacturer discovers that if he varies the price of an item a little bit up or down from $3 he'll make less profit than if he charges $3 exactly. He might conclude that $3 represents a local maximum for profit, and that's a reasonable conclusion. So how much should he charge? Three dollars? Not necessarily. It might be that if he increased the price enough then sales revenue might start to increase again. Sometimes people avoid buying something if it appears too cheap. It might well be that if he priced the item at $9.99 he'll make more profit than if he charged any other price. This, then, should be the price he should charge.
Sometimes you have to go down from a maximum in order to go up even higher. Every mountain climber knows that if we want to scale the highest peak we'll have to spend a certain amount of time going up and down smaller peaks.
The GLOBAL MAXIMUM (MINIMUM) of f(x) for x in a certain range of values, is the LARGEST (SMALLEST) value of f(x) for x in that range.
A local maximum need not be a global one. The curve could go even higher at another stage. A global maximum is very often one of the local maxima, but not always. It could be an endpoint of the range of values.
How to Locate the Global Maximum and Global Minimum (where they exist).
(1) Check whether a global maximum or minimum exists.
(2) Find the stationary points.
(3) Find the end-points.
(4) Find any points where the derivative doesn't exist.
(5) Evaluate the function at each of these points.
(6) Select the smallest and largest of these values.
Maxima and Minima for Functions of Two Variables
At the top of a hill it doesn't matter in which direction you walk - it's flat in every direction, for an instant. The tangent plane is horizontal and so all the tangents have zero slope. The same holds at a local minimum. So to find local maxima and minima for a function of two variables z = f(x, y) you look for points where ∂z/∂x = 0 and ∂z/∂y = 0. These points are called stationary points.
A stationary point for a function z = f(x, y) is a point where ∂z/∂x = 0 and ∂z/∂y = 0.
