Discussion of Extreme Values of Continuous Functions
Extreme Value Theorem
The extreme value theorem states that a function f has an absolute maximum and an absolute minimum on any closed bounded interval [a,b] such that f is continuous on [a,b].
We also have the
Critical Number Theorem
If a continuous function f has a relative extremum at c, then c must be a critical number of f.
where the definition of a critical number is a number 'c' such that f '(c) = 0, or the derivative doesn't exist at c.
Together these two theorems give us a very simple process to follow in order to find the absolute minimum and absolute maximum of a continuous function on a closed interval.
We first compute f '(x) and find all of the critical numbers in the interval [a,b]. We then make a list of all the critical numbers and the endpoints a, b. We evaluate the function f at each of these points. Finally, we compare all of the function values. The largest value of f is the absolute maximum of f on [a,b] and the smallest value of f is the absolute minimum of f on [a,b].
Example 1
f(x) = 10 + 6x - x 2 on [-4, 4]
First find f '(x) = 6 - 2x. Note f '(x) is defined everywhere, so we don't have any troubles there. Next, we set f '(x) = 0 = 6 - 2x, so f '(3) = 0. We can then make a table:
Comparing values, we see that the absolute maximum is 19 which occurs when x = 3 and the absolute minimum is -30 which occurs when x = -4.
Example 2
f(x) = |x - 4| on [-1, 5]
First we notice that there is a corner in the graph when x = 4, so f '(4) is not defined, so 4 is a critical number. Also, the derivative is never zero. So, we make our table:
Comparing values, we see that the absolute maximum is 5 which occurs when x = -1 and the absolute minimum is 0 which occurs when x = 4.
Final Comment
I would like to note that the above procedure only works when f is continuous on a closed interval [a,b]. Therefore, if there are any discontinuities in your function, you also need to include those points of discontinuity in your table.