Find the critical numbers of f (if any). Find the open...

Find the critical numbers of $f$ (if any). Find the open intervals on which the function is increasing or decreasing and locate all relative extrema. Use a graphing utility to confirm your results.
$$
f(x)=5-|x-5|
$$

Key Concepts

Critical Numbers

Critical numbers are points in the domain of a function where its derivative is zero or undefined. These points are important because they are the possible locations of relative extreme values (local maxima and minima) or points of inflection, and they provide insights into the overall behavior of a function.

Absolute Value Function

The absolute value function converts negative inputs into their positive counterparts and is typically defined piecewise. Its piecewise nature can lead to different derivative behaviors on different intervals and may result in non-differentiability at the point where the expression inside the absolute value becomes zero, which is crucial for analyzing critical points.

Increasing/Decreasing Intervals

Increasing and decreasing intervals are determined by the sign of the function's derivative. When the derivative is positive on an interval, the function is increasing there, and when it is negative, the function is decreasing. Identifying these intervals is essential for understanding the function's behavior and for locating relative extrema.

Relative Extrema

Relative extrema refer to the local maximum or minimum values of a function. They occur at critical numbers where the function transitions from increasing to decreasing (or vice versa). The identification of these extrema provides valuable information for analyzing and graphing the function.

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