. Likewise, there exists at least one point d at which f(x) achieves its minimum value. That is, for all x in the interval [a,b] we have that 
.Here is a theorem about the behavior of continuous functions. The proof of this theorem is beyond the scope of this course, so I will simply state the theorem without proving it.
Extreme Value Theorem Let f(x) be a continuous function on the closed interval [a,b]. There exists at least one point c in the interval [a,b] at which the function achieves its maximum value. That is, for all x in the interval [a,b] we have that . Likewise, there exists at least one point d at which f(x) achieves its minimum value. That is, for all x in the interval [a,b] we have that .
This theorem is an example of what mathematians call an existence theorem. The theorem tells us that the maximum and minimum values exist, but does not tell us where or how to find them. For that we will need to develop some additional tools.
A point x = c is called a local maximum for the function f(x) if f(c) is greater than or equal to f(x) for all of the x values near c. To make this definition a little more concrete, we have to say what it means for x to be near c.
Definition (Local Maximum) We say that the function f(x) has a local maximum at x = c if there is an interval (a,b) containing c such that for all x in (a,b) we have 
.
As a special case of this definition if the interval (a,b) is actually 
we say that c is a global maximum as well.
With this definition in place, we are now prepared to write down our first useful theorem.
Theorem If f(x) is differentiable at x = c and takes on a local maximum value at x = c, then 
Proof Suppose that f(x) has a maximum value at c. Look at its derivative at x = c:
| (1) |
Note that f(x) takes its maximum value at c. For all other x nearby, 
. This tells us that
From this we see that the numerator is either negative or possibly 0. To compute the limit (1), we can compute the limits from either direction.
| (2) |
| (3) |
Remember, f(x) is differentiable at c. Therefore the limit (1) must exist. The only way to reconcile the limits (2) and (3) with each other is to say that the limit has to be 0.
The maximum value theorem is very frequently misused, so it is worth our while to spend a little time looking at it more closely and seeing how to use it correctly. In condensed form this theorem says
We get the contrapositive form for free:
or
The thing that most often causes trouble is that people will try to use the converse of this theorem:
This latter statement is not true in general.
As a simple counter-example, consider f(x) = x3 at x = 0. Although 
is 0 when x is 0, f does not have a maximum at x = 0.
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Despite the restrictions on its use, the theorem is still very useful. The theorem says that all maximum points of the function have derivative equal to 0. Thus, if you want to find where the maxima are, check out the places where the derivative is 0. Those places are called critical points. A critical point is not automatically a maximum point, but since all maximum points are critical points, the critical points are the best place to start your search for maximum points.
A similar theorem holds for minima:
Theorem If f is differentiable at x = c and takes on a local minimum value at x = c (that is, 
for all x near c), then 
We have established that extreme points are critical points, so we know to go looking for critical points. The problem is that once we have found a critical point we do not know exactly what to do with it. There are two strategies we can use to solve this problem.
Value test Compare the value of the function with nearby values. If the critical point is, say, a maximum, the value of the function at that point will be greater than the value at nearby points.
First derivative test If the critical point is a maximum, the function should be increasing (positive derivative) just to the left of that point and decreasing (negative derivative) just to the right.
Here is an example. Consider the function
That function has derivative
Which is 0 at x = 0 and x = 1. These are the critical points. To see whether or not these critical points are extreme points, we can look at the behavior of the derivative near these critical points. Here is what the derivative does at a number of points near 0 and 1.
| x |  |
|---|---|
 |  |
 |  |
 |  |
This shows that as x approaches 0, the derivative is negative, goes to 0 and then becomes negative again. This indicates that the point x = 0 is neither a maximum nor a minimum point. As we pass through 1, we see a function that is decreasing, flattens out, and then increases again. This indicates clearly that the function has a minimum value at x = 1. If we plot the graph we can see that this is indeed the case.
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The second derivative also provides useful shape information for functions. To see this, lets look at an example of a function which has a positive second derivative everywhere.
As you can see from a plot of this function, it is bowl shaped with the bowl set upright.
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The bowl shape here goes along with the second derivative being positive. The shape is called concave upward (or we say that the function is concave up). A simple modification of the last example will give us an example of a function whose second derivative is negative everywhere.
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This shape, which goes along with negative second derivative, is called concave down.
Most functions alternate between concave up and concave down. Consider the example
The second derivative is positive over part of the x-axis. That part is given by
6 x - 4 > 0
6 x > 4
Likewise, the second derivative is negative over the remainder of the x-axis.
6 x - 4 < 0
6 x < 4
The point where the second derivative changes from being positive to negative is called an inflection point. To find it, look for
In this example, the single inflection point is located at x = 2/3. If we examine a plot of this function we can clearly see the concave and concave down portions of the graph.
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We can summarize what we have learned here by saying
Theorem Suppose f(x) is differentiable everywhere on an open interval (a,b). If x = c is a critical point of the function f(x) located somewhere in the interval (a,b) and 
, then c is a local maximum for f(x). Likewise, if 
, then c is a local minimum for f(x).
Let us use what we just learned about critical points to construct a graph of the function
The critical points are located at
or
To determine the exact nature of these critical points, we can use any of the three methods above.
The first method is to make a table of the value of the function f(x) at these points and at points nearby:
| x |  |
|---|---|
| -0.8 | 1.31 |
 | 1.32 |
| -0.6 | 1.28 |
| -0.1 | 1.01 |
| 0 | 1 |
| 0.1 | 1.01 |
This information shows that the critical point 
is a local maximum, and that the critical point x = 0 is a local minimum value.
The second method is to examine the behavior of the first derivative near the critical points.
| x |  |
|---|---|
| -0.8 | 0.448 |
 | 0 |
| -0.6 | -0.552 |
| -0.1 | -0.1995 |
| 0 | 0 |
| 0.1 | 0.2005 |
The third method is to compute the second derivative at the critical points.
This shows that the function is concave upward at 0 (which makes a local minimum there) and concave downward at 
(which makes a local maximum there).
The plot confirms all of this:
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32, 33, 41, 42, 52, 53, 70, 73
