The function f(x)=|(x^2-2)/(x^2-4)| has (A) no point of local minima (B) no point of local maxim

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The function f(x)=|(x^2-2)/(x^2-4)| has (A) no point of...

The function $\mathrm{f}(\mathrm{x})=\left|\frac{\mathrm{x}^{2}-2}{\mathrm{x}^{2}-4}\right|$ has
(A) no point of local minima
(B) no point of local maxima
(C) exactly one point of local minima
(D) exactly one point of local maxima

Key Concepts

Critical Points

Critical points are values in the domain of a function where the derivative is zero or undefined. These points are key in the process of identifying potential local minima or maxima since they represent changes in the behavior of the function's slope.

Domain Analysis

Domain analysis entails determining the set of all input values for which a function is defined. For rational functions, this often involves excluding values that cause the denominator to be zero, which is essential for accurately investigating the overall behavior, especially when searching for local extrema.

Rational Functions

Rational functions are ratios of polynomial functions. Their analysis includes determining the domain (by finding values that make the denominator zero), understanding asymptotic behavior, and investigating continuity. These aspects are crucial when examining features such as local extrema and discontinuities.

Absolute Value Transformation

This concept refers to applying the absolute value operation to a function, which converts every output to a nonnegative value. It can alter the differentiability and the overall shape of the graph by reflecting negative portions above the x?axis and creating potential points (or corners) where the derivative might not exist.

Local Extrema

Local extrema involve finding the points on a function where the values are locally highest (maxima) or lowest (minima) compared to surrounding points. Determining local extrema generally involves examining the derivative, testing critical points, and considering the impact of any transformations (like the absolute value) on the behavior of the function.

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