Maximize Function Values With The Closed Interval Method: A Guide To Find Absolute Extrema On Closed Intervals

closed interval method (for finding absolute extrema on [a,b]

1. evaluate f at each critical point2. evaluate the endpoints, f(a) and f(b)3. the largest value is the absolute maximum value. the smallest value is the absolute minimum value

The closed interval method is a technique used to find the absolute extrema of a continuous function on a closed interval [a,b]. The method involves the following steps:

1. Find the critical points of the function on the interval [a,b]. These are the points where the derivative of the function is zero or undefined.

2. Evaluate the function at the critical points and the endpoints a and b. This gives a set of candidate points for the absolute extrema of the function.

3. Compare the values of the function at the candidate points to determine the absolute maximum and minimum values.

4. Check that the absolute maximum and minimum values are indeed attained at the candidate points. This can be done by either showing that the function is monotonic in each subinterval between adjacent critical points, or by showing that the endpoints a and b are not themselves local extrema.

It’s important to note that the closed interval method assumes that the function is continuous on the interval [a,b]. If the function is discontinuous on the interval, then the method cannot be used to find the absolute extrema.

Overall, the closed interval method is a useful tool for finding the absolute extrema of a continuous function on a closed interval. By identifying critical points and evaluating the function at a finite set of candidate points, we can determine the global maximum and minimum values of the function on the interval.

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