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Critical points are values in the domain of a function where its derivative is either zero or undefined. These points are crucial when analyzing the behavior of a function because they can indicate potential areas where the function might have a local maximum, local minimum, or an inflection point. For example, in identifying the critical points for a function like \( g(x) = x^3 - 3x \), one would calculate the derivative and set it equal to zero: \( g'(x) = 3x^2 - 3 \). When solved, this yields \( x = \text{±}1 \), indicating that these are critical points where the function's slope changes and further investigation is required to determine the nature of these points.

The Second Derivative Test is a method used to determine the concavity of a function at its critical points and thus help classify whether these points are local maxima, local minima, or points of inflection. This test involves taking the second derivative of the function and then evaluating it at the critical points. For instance, let's look back at our function \( g(x) \).

We found that the second derivative of \( g(x) \) is \( g''(x) = 6x \). By substituting the critical points into \( g''(x) \), we can infer that at \( x = -1 \), the function has a local maximum since \( g''(-1) = -6 \), and at \( x = 1 \), it has a local minimum since \( g''(1) = 6 \). A negative second derivative means the graph is concave down at that point, indicating a local maximum, while a positive one implies concave up, indicating a local minimum.

Absolute extrema of a function refer to the highest and lowest values that the function attains over its entire domain, also known as the global maximum and minimum. In some cases, as with the function \( g(x) = x^3 - 3x \), there may not be any absolute extrema. This particularly occurs with polynomial functions of odd degree which are not bounded above or below, meaning they increase or decrease without bound as \( x \) approaches positive or negative infinity. To formally prove that a function does not have any absolute extrema, one would demonstrate that for any given value, it is possible to find another value in the domain where the function takes on an even higher or lower value, confirming the absence of an absolute maximum or minimum.

Local maximum and minimum points refer to locations on the graph of a function where the function value is greater than (maximum) or less than (minimum) the values of the function at points nearby. These are also sometimes called relative extrema because they are not necessarily the highest or lowest points on the entire graph, just in a localized area. Using our function \( g(x) = x^3 - 3x \) as an example, we can determine that \( x = -1 \) is a local maximum since \( g(-1) = 2 \), and \( x = 1 \) is a local minimum since \( g(1) = -2 \). These points are visually apparent on the graph, as they are the peaks and valleys in the immediate vicinity of the point.