A. Locate the critical points of f b. Use the First...

a. Locate the critical points of $f$
b. Use the First Derivative Test to locate the local maximum and minimum values.
c. Identify the absolute maximum and minimum values of the function on the given interval (when they exist).
$$f(x)=\frac{x^{2}}{x^{2}-1} ;[-4,4]$$

Key Concepts

Rational Functions and Domain Considerations

In the context of rational functions, it is essential to consider the domain restrictions arising from the denominator. Points where the denominator is zero are not in the domain of the function and must be excluded from the analysis of critical points. This consideration is particularly important when the interval of interest potentially includes such points.

Absolute Extremum on a Closed Interval

When determining absolute (global) maximum and minimum values of a continuous function on a closed interval, one evaluates the function at its critical points that lie within the interval and at the endpoints of that interval. The largest and smallest values obtained from this evaluation represent the absolute maximum and minimum values, respectively.

First Derivative Test

The First Derivative Test is a method used to determine whether a function's critical point is a local maximum, a local minimum, or neither. By analyzing the sign changes of the derivative before and after the critical points, one can infer the nature of the extremum. Specifically, a sign change from positive to negative indicates a local maximum, while a change from negative to positive indicates a local minimum.

Critical Points

Critical points are values in the domain of a function where the derivative is zero or undefined. These points are significant because they are potential locations for local extrema. In the analysis of functions, one must first identify these points before further classifying them using tests like the First Derivative Test.

Local Extrema

Local extrema refer to the local maximum and minimum values of a function. These are the values of the function at points where it reaches a peak (local maximum) or a trough (local minimum) within a certain interval, determined by the behavior of the function in the immediate vicinity of those points.

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