Provided some conditions are satisfied, the Extreme Value Therom guarantees the
existence of maximum and minimum values. But how am I supposed to find those
maximum and minimum values? We've actually already done this in some
examples, but it's worth describing an explicit process for finding maxima and
minima. Here's a frour-step process.
First, differentiate your function Can find all the critical points.
Those are places where the derivative is equal to zero or the derivative isn't
defined. And also lists the end points if they're
included in your domain. Check those points, right?
Check the end points, check the critical points, and potentially you also need to
check the limiting behavior if you're working on an open Interval.
Let's work an example. Okay, let's work an example.
Let's look at the function, given by the rule f of x equals 1 over, x squared minus
1. This quantity squared.
But let's only consider this function on a restricted domain.
Let's consider, a maximum, minimum values of this function on the interval between
minus 1 and 1. This open interval.
And now I differentiate. Yeah, the first step is to differentiate
this function. So before I differentiate it, I'm just
going to rewrite this. Instead of 1 over x squared minus 1
squared, I'm going to write this as x squared minus 1 to the negative 2nd power.
It's going to make it a little bit easier for me when I think about the derivative.
So what's the derivative of this function? By the power rule and the chain rule This
is negative 2 times the inside, x squared minus 1 to the negative 3rd power times
the derivative of the inside which is 2x or another way to write this is negative 2
times 2x, that's minus 4x divided by x squared minus 1 to the third power.
With the derivative in hand, I can find the critical points.
Yeah, the next step is to list the critical points, and also the end points
if they're included. So let's see.
What are the critical points of this function?
Will those be places where the derivative doesn't exist, or the derivative is equal
to zero. Let's first think about when the
derivative is equal to zero. Well, in order for that derivative to be
equal to zero, it's a fraction. When's a fraction equal to zero?
Well, that occurred exactly when the numerator is equal to zero.
So I'm really just asking when is minus 4x equal to zero?
And that's just when x is equal to 0. So the only time when the derivative is
equal to 0 is when x is equal to 0. Now the derivative doesn't exist when x is
equal to 1 or when x is equal to minus 1 but, you know, the functions not even
defined in that case anyway. And I'm only considering numbers between
minus 1 and 1 anyhow. Don't have to worry about those.
And I don't have to worry about end points.
Because, again, the interval that I'm considering doesn't include the end
points, It's an open interval. So I'm just going to check x equals zero
to figure out what's going on there. So let's think about this point where x is
equal to zero. Well the derivative at zero is equal to
zero. If I just plug in zero into this I can see
that. What's the derivative at a number between
minus 1 and zero? Well if I plug in a number between minus 1
and zero, x squared is then between zero and 1, so x squared minus 1 is now a
negative number cubed. The denominator is negative, but this is a
negative number, negative 4 times a negative number.
So we've got negative times negative divided by negative, that's negative.
So the derivative at a number between minus 1 and 0 is negative in this case.
And a similar kind of reasoning will show that the derivative is positive if I
evaluate the derivative between 0 and 1. So here is the deal.
I've got one critical point on this interval between minus 1 and 1.
And the function's decreasing between minus 1 and 0, and it's increasing between
0 and 1. So what does that mean about the point 0?
Well it means that the point 0 must be a place where the function achieves a local
minimum value. So I can summarize that here, f of 0.
Is by, this is the first derivative test is a local minimum value.
What about the end behavior? What happens when x is near negative 1, or
when x is near 1? So that's really step 4 in this checklist,
is to check the limiting or the end behavior of this function.
What do I mean by that? I really mean look at this when X is just
a little bit bigger than minus Y, or X is a little bit less than 1.
Alright, that's the end behavior of this function.
These end points are actually included so of course I can't evaluate the function
there but I should still see what happens when X is close to these endpoints.
Let's imagine I was trying to draw a graph of the function, alright.
How would I know from the derivative. I know the function's decreasing here and
increasing here. And I know the function's got a local
minimum value here, but linking to the first derivative test.
So the function's going down and then it's going up.
But what happens at these two Values, when I get close to minus 1 and when I get
close to 1. When I get close to minus 1, the function
is very, very large, and when I get close to 1, on this side, the function's also
very, very large. So if I imagined trying to draw a graph of
this function? Right, it would be coming down like this,
flattening out and then it would be getting very, very large again.
Let's summarize the situation. This graph will help us summarize the
situation. Alright, what do I know.
I know that F of 0 is the smallest output, it's the minimum value that the function
achieves on the domain, open interval minus 1 to 1.
And there's no maximum value. By considering this limiting behavior, as
x approached to the endpoints which aren't included in this domain.
I can see that this function's output can be made as positive as I'd like, right.
I can choose values of X to make the output as big as I want.
So I can't say this function actually achieves some maximum output value.
It doen'st achieve some maximum output value.
But it does achieve a minimum output value right here.
When x is equal to 0. And that output is 1.
It's important to emphasize that the domain truly matters.
Yeah, what if I ask to find the maximum and minimum values of the same function f
of x is equal to one over x squared minus one squared, but a different domain?What
if I looked on the interval from one to infinity this open interval?
Again we start by differentiating Well of course, the derivative's the same.
The derivative of this function is still minus 4 x over x squared minus 1 to the
third power. Now, where are the critical points?
The only place that this derivative is equal to 0, is when x is equal to 0.
And the only place where the derivative doesn't exist is just places where the
function isn't even defined, right? So on this interval.
From one to infinity, there's no place where the derivative isn't defined or is
equal to zero. So there's no critical points.
So what about the behavior near the boundaries of this interval?
What happens if X is just a little bit more than one or when X is really, really
big? So, we working on the interval from 1 to
infinity. This open interval.
So, that means the limits that you'd consider are the limits, the function as x
approaches infinity and the limit of the function as x approaches 1 from the right
hand side. And, doing calculation I find that the
limit of the function as x approaches infinity is 0 because, in that case the
denominator can be made as large as I like, which makes this ratio as close to 0
as I like. And, the limit of the function as x
approaches 1 from the right is equal to infinity.
Because in that case if x approaches 1 from the right this denominator can be
made as close to 0 and positive as I'd like which makes this ratio as large as
I'd like. So in this case this function on this
domain doesn't take a largest value and doesn't take a smallest value.
The function's output can be made as big as I'd like if I'm willing to choose x
just a little bit bigger than 1 and the function's output can be made as close to
0 as I'd like if I'm willing to choose X to be big enough.
In other words, the domain matters. I mean this function, if I'm considering
the funciton on this domain, from -1 to 1. It does have a smallest output value.
And it should use its local minimum value when x is equal to 0.
But if consider the same function. But on a different domain there's no
critical point on this domain, right? And I studied the limiting behavior as x
approached one, and as x approached infinity, and I found that I could make
the output of this function as large as I'd like and as close to zero as I'd like.
So this function doesn't achieve a maximum minimum value on this interval.