Maxima and Minima of Functions
In many problems, we want to know just how big or small a function can get. We call the points where these values are reached the maxima and minima of the function. For example, you might want to know the minimum time it takes for a brussels sprout to decompose on the compost heap. Will it decompose before your parents find out what you've been up to?
Often, we can only see a small part of the graph of a function and need to work out its maximum and minimum values for that part of the function. This is a bit like putting the function under a microscope.
Local Maxima and Minima
The maximum and minimum value in the small part of the graph we can see may not be the maximum or minimum for the whole function. We call maxima (the plural of maximum) or minima (the plural of minimum) of this sort local maxima or minima. On the graph of the function they look like the tips of mountains or the bottoms of valleys. They might be nicely rounded or pointy. Local maxima and minima are easy to spot when you look at graphs:
Here are some nicely rounded maxima and minima. It's easy to see where they are: there are two local maxima and one local minimum in the above picture.
And here are some pointy local maxima and minima: they are even easier to spot. There are two local maxima, and one local minimum.
As you can see, it's easy enough to spot local maxima and minima. However, mathematicians are picky people – it isn't enough to just say you can see things. They want you to know how to define them.
So how do we define a Local Maximum?
You need to start by deciding what "local" means. How about choosing an interval as shown in the diagram below and looking for the largest value of
in that interval?
Let's try to write a definition that will satisfy a mathematician:
local maximum is a point \(a\) at which the height of the function-value \(f(a) \)
is greater than or equal to the height of the function at \(x-values\) taken from anywhere in an interval.
Now let's make the mathematicians even happier by including some symbols:
If \(f(a) \geq f(x)\) for all \( x \) in an interval, then the function has a local maximum at the point \( (a, f(a)) \).
Note: In this definition, we want to lie somewhere in the middle of the interval, not at one of its endpoints.
A function can have more than one local maximum – the one in our picture has two!
What about a Local Minimum
We can come up with a similar definition for a
local minimum just by switching the inequality signs around. How's that for laziness?
Here's our new definition:
If \( f(a) \leq f(b) \) for all \( x \) in an interval, then the function has a local minimum at the point \( (a,f(a)) \).
Can there be more than one local minimum? Of course! A graph can have two or more valleys.
If we want to talk about the local minima and local maxima of a function together, we call them local extrema.
Global (or Absolute) Maximum and Minimum
If you want to find the largest or smallest value of the entire function, then you are looking for an "Absolute" or "Global" maximum or minimum.
There can only be one global maximum and one global minimum. Some functions have no global maximum or minimum.
If you look at the function in the picture we started with, and assume that it continues downwards on the left and right, you can see that this function has one global maximum and no global minimum:
Some people might say that the function has a global minimum at \( - \infty \), but this really just means that it goes on and on forever, never reaching a minimum value.
Tracking down the exact maximum and minimum values
You might be able to get an approximate maximum or minimum value for a function if you draw its graph carefully on graph paper. If you want to get exact maximum and minimum values, you'll need to use calculus – derivatives will do the job for you.