Use the given derivative to find all critical points of f...

Use the given derivative to find all critical points of $f$ and at each critical point determine whether a relative maximum, relative minimum, or neither occurs. Assume in each case that $f$ is continuous everywhere.
$$
f^{\prime}(x)=\ln \left(\frac{2}{1+x^{2}}\right)
$$

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Key Concepts

Critical Points

Critical points are values in the domain of a function where its derivative equals zero or does not exist. These points are important because they are potential locations of local maxima, local minima, or inflection points.

First Derivative Test

The first derivative test involves checking the sign of the derivative before and after a critical point. If the derivative changes from positive to negative at a critical point, it indicates a local maximum. Conversely, a change from negative to positive suggests a local minimum. If no sign change occurs, the critical point is not an extremum.

Sign Analysis of the Derivative

Sign analysis of the derivative is used to determine intervals where the function is increasing or decreasing. By assessing the sign of the derivative across different regions, one can predict the behavior of the function and classify the critical points accordingly.

Function Continuity

Continuity of a function ensures that there are no breaks, jumps, or holes, making the analysis of critical points through derivative tests reliable. If a function is continuous, the results from derivative-based methods like the first derivative test are valid and complete.

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