Extreme Value Theorem Implicit Differentiation Extreme Value TheoremExtreme Value Theorem Extreme Value Theorem

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. Implicit Differentiation Extreme Value TheoremExtreme Value Theorem Extreme Value Theorem

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

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

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? Implicit Differentiation Extreme Value TheoremExtreme Value Theorem Extreme Value Theorem

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? Implicit Differentiation Extreme Value TheoremExtreme Value Theorem Extreme Value Theorem

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

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. Implicit Differentiation Extreme Value TheoremExtreme Value Theorem Extreme Value Theorem

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. Implicit Differentiation Extreme Value TheoremExtreme Value Theorem Extreme Value Theorem

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

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