Extreme Value Theorem
We know. The title of this reading sounds pretty gnarly. The extreme value theorem, though, is just a slight extension of the boundedness theorem. There's really nothing all that extreme about it.
Maximum and Minimum Values
The maximum value of a function on an interval is the largest value the function takes on within that interval. Similarly, the minimum value of a function on an interval is the smallest value the function takes on within that interval:
A function may hit its maximum and/or minimum value on an interval more than once. The function f(x) = sin(x) on the interval [-2π, 2π] hits its maximum and minimum two times each:
Look again at the function f(x) = sin(x) on [-2π,2π]:
The maximum value of the function on this interval is 1. The function attains its maximum at and at . The minimum value of the function on this interval is -1. The function attains its minimum at and at .
Graph the function f(x) = 1 on the interval [0,1]:
The largest value this function hits on the interval is 1, therefore its maximum value is 1. The smallest value this function hits on the interval is 1, so its minimum value is also 1. The function attains both its maximum and its minimum value at every value of x in the interval. How's that for weird?
Now we're ready to relate the idea of maximums and minimums to continuous functions.
Extreme Value Theorem: A function f that is continuous on a closed interval [a, b] must attain a maximum and a minimum on that interval.
To see why this is different from boundedness, look at this function:
This function is bounded, but it never actually reaches a maximum or minimum value. As x approaches ∞ the function is always increasing, approaching N but never quite reaching N. As x approaches -∞ the function is shrinking, approaching M but never quiten reaching it.
Since a function can be bounded without hitting a maximum or minimum value, the Extreme Value Theorem does say something different from the Boundedness Theorem. In fact, the Extreme Value Theorem is actually a stronger theorem. What we mean by this is that the Boundedness Theorem is really a special case of the Extreme Value Theorem, since a function that attains a maximum and minimum value on a closed interval is also bounded by those values.
To see why the Extreme Value Theorem makes sense, we'll draw some functions. If we have a continuous function on a closed interval [a,b], it must hit its maximum value in the interval or at an endpoint, as it does here:
Similarly, the function must hit its minimum value either in the middle of the interval, or at an endpoint.
The Extreme Value Theorem makes the same two assumptions as the Boundedness Theorem, but draws a slightly different conclusion. If we assume that both
- f is continuous on an interval, and
- that interval is a closed interval [a,b]
then we can conclude that f hits a minimum and a maximum value on that closed interval.
Again, if either assumption is missing, we're not allowed to draw the conclusion. If f is discontinuous on [a, b], then f might not hit a maximum or minimum value:
If f is continuous on an interval but the interval is not closed, then f might not hit a maximum or minimum value:
Both assumptions are absolutely necessary.