Extreme Value Theorem

1. Extreme Value Theorem

2. Objectives Students will be able to • Find all absolute maximum and minimum points of a function on a closed interval. • Determine if the Extreme Value Theorem applies to a given situation.

3. Extreme Value Theorem Suppose f is a continuous function over a closed bounded interval [a, b], then there exists a point d in [a, b] where f has a minimum and a point c in [a, b] where f has a maximum, so that for all x in [a, b].

4. In summary, to find absolute extrema, we need to look at the following types of points: i. Interior point in the interior of the interval [a, b] where f’ (x) = 0 ii. End points of [a, b]. We find all function values for the x-values found in i and ii. The largest function value us the absolute maximum. The smallest function value is the absolute minimum on the interval [a, b].

5. Example 1 For the graph shown, identify each x-value at which any absolute extreme value occurs. Is your answer consistent with the Extreme Value Theorem?

6. Example 2 For the graph shown, identify each x-value at which any absolute extreme value occurs. Is your answer consistent with the Extreme Value Theorem?

7. Example 3 Label each point from the function graphed in example 1 as an absolute maximum, absolute minimum, or neither.

8. Example 4 Find the absolute maximum and minimum values of the function over the interval [―1, 0] and indicate the x-values at which they occur.

9. Example 5 Find the absolute maximum and minimum values of the function over the interval [―3, 3] and indicate the x-values at which they occur.

10. Example 6 Find the absolute maximum and minimum values of the function over the interval [―1, 4] and indicate the x-values at which they occur.

11. Example 7 Find the minimum value of the average cost for the cost function on the interval .