Identifying and Locating Relative Extrema | A Comprehensive Guide in Mathematics

Relative (local) max or min occurs at…

In mathematics, a relative (local) maximum or minimum occurs at a point on a graph where the function reaches its highest or lowest value within a specific interval

In mathematics, a relative (local) maximum or minimum occurs at a point on a graph where the function reaches its highest or lowest value within a specific interval. Relative extrema are distinct from absolute extrema, which refer to the highest or lowest points on the entire graph of a function.

To determine the presence and location of relative extrema, we need to follow these steps:

1. Find the critical points: Critical points occur when the derivative of the function equals zero or is undefined. These points can potentially be candidates for relative extrema. To find critical points, we set the derivative of the function equal to zero and solve for the values of x.

2. Check the endpoint values: If the function is defined by an interval or domain with defined endpoints, we need to evaluate the function at these endpoints as well.

3. Apply the first or second derivative test: Once we have identified the critical points and evaluated the function at the endpoints, we can use the first or second derivative test to determine whether each critical point corresponds to a relative maximum or minimum.

– The first derivative test involves analyzing the sign changes in the derivative function around each critical point. If the derivative changes from positive to negative, then we have a relative maximum. Conversely, if the derivative changes from negative to positive, we have a relative minimum.

– The second derivative test involves calculating the second derivative of the function at each critical point. If the second derivative is positive, then we have a relative minimum. If the second derivative is negative, then we have a relative maximum. If the second derivative is zero, the test is inconclusive.

By following these steps, we can identify the presence and location of relative extrema on a graph.

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