Step-By-Step Guide to Finding Absolute Extrema in Mathematics

Find absolute extrema

To find the absolute extrema of a function, we need to identify the highest and lowest points on a specific interval

To find the absolute extrema of a function, we need to identify the highest and lowest points on a specific interval. Here is a step-by-step guide to finding the absolute extrema:

1. Identify the domain: Determine the interval over which you want to find the absolute extrema. This could be a specific range of values or the entire domain of the function.

2. Find critical points: Locate the points where the derivative of the function equals zero or where the derivative is undefined. To find critical points, set the derivative of the function equal to zero and solve for x. If the derivative is undefined at any point, check if it corresponds to a maxima or minima.

3. Evaluate the function at critical points: Substitute the critical points in the original function and calculate the corresponding y-values.

4. Evaluate the function at the endpoints: If the interval defined in step 1 has endpoints, substitute these values into the original function and calculate the corresponding y-values.

5. Compare the values: Compare the values obtained in steps 3 and 4 to determine the lowest and highest points on the given interval. The lowest point is the absolute minimum, and the highest point is the absolute maximum.

Note that there may be multiple absolute maxima or minima on a given interval. Additionally, if the domain of the function is unbounded, there may not be absolute extrema.

It’s important to keep in mind that this process applies to continuous functions on a closed and bounded interval, according to the Extreme Value Theorem. For functions that are neither continuous nor on a closed interval, further analysis is required.

Overall, finding the absolute extrema involves identifying critical points and evaluating the function at those points and the endpoints of the interval.

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