1. Extreme Value Theorem Implicit Differentiation
Extreme Value TheoremExtreme Value Theorem
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.
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Extreme Value TheoremExtreme Value Theorem
Extreme Value Theorem

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].
Implicit Differentiation
Extreme Value TheoremExtreme Value Theorem
Extreme Value Theorem

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].
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Extreme Value TheoremExtreme Value Theorem
Extreme Value Theorem

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?
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Extreme Value TheoremExtreme Value Theorem
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?
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Extreme Value Theorem

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

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.
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Extreme Value Theorem

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.
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Extreme Value TheoremExtreme Value Theorem
Extreme Value Theorem

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.
Implicit Differentiation
Extreme Value TheoremExtreme Value Theorem
Extreme Value Theorem

11. Example 7
Find the minimum value of the average cost for the cost function
on the interval
Implicit Differentiation
Extreme Value TheoremExtreme Value Theorem
Extreme Value Theorem