Mastering the search for relative extrema: A guide to finding maximum and minimum points within a specified interval

Relative extrema

Either a relative minimum or relative maximum

Relative extrema can be defined as the maximum and minimum points of a function within a specific interval. These points are known as relative because they are only compared to the values of the function within a small neighborhood or interval around them. The values of the function outside this neighborhood may be higher or lower than the relative extrema.

To find the relative extrema of a function, we need to find the critical points. Critical points are the points at which the derivative of the function is zero or undefined. We need to check both the left and right-hand limits of the function at these critical points to determine whether they correspond to a relative maximum or minimum.

If the left-hand limit is greater than the right-hand limit at a critical point, the function has a relative maximum at that point. Conversely, if the right-hand limit is greater than the left-hand limit, the function has a relative minimum at that point. If both limits are equal, then the critical point may be an inflection point, a point of horizontal inflection, or a saddle point.

For functions that are continuous and differentiable, the First Derivative Test can be used to determine whether a critical point is a relative maximum or minimum. The Second Derivative Test can be used when the first derivative test is inconclusive.

In summary, relative extrema are the maximum and minimum points of a function within a specific interval. To find them, we need to find the critical points of the function and evaluate the function at a small neighborhood around those points to ascertain whether they correspond to a relative maximum or minimum.

More Answers: