Closed Interval Method
In fact when we used derivative to find extrema, this extrema mainly were local.
Now we turn our attention to finding global extrema on closed interval .
Extreme Value Theorem. If is continuous on a closed interval , then attains a global maximum value and a global minimum value at some numbers and from interval .
Extreme values can be taken either at endpoints of interval or inside it. On the figure below one case is when both extrema are taken at endpoints and another case when both extrema are taken inside interval. Of course there can be cases when one extremum is at endpoint and another is inside interval.
Note that extreme value can be taken more than once.
This theorem doesn't hold when either interval is not closed or function is not continuous.
For example, consider function on interval . It is discontinuous at . Since and then function doesn't attain neither global maximum, nor global minimum.
Now consider function on interval (note that interval is not closed). It is continuous on this interval and decreasing. Since then function grows without a bound and so there is no global maximum. Since then function approaches minimum value , but never attains it! So, there are no global minimum as well.
Although Extreme Value Theorem states that for continuous function on closed interval there are global maxima and minima but it doesn't tell us how to find them.
Luckily, we already know how to find local extrema and on closed interval global extrema occurs either at stationary points or on endpoints of interval.
Closed Interval Method. To find the global maximum and minimum values of a continuous function on a closed interval we need to do following three steps:
- Find the values of at the stationary points of in .
- Find the values of at the endpoints of the interval.
- The largest of the values from Steps 1 and 2 is the global maximum value; the smallest of these values is the global minimum value.
Example. Find global extrema of on interval .
We have that . when and .
There are no such that doesn't exist.
Therefore stationary points are and .
Now find values of function at stationary points:
and .
Calculate values of at the endpoints:
and .
So, the largest value is - this is global maximum; and the smallest value is - this is global minimum.