Find the intervals on which f increases and the intervals on...

Find the intervals on which $f$ increases and the intervals on which $f$ decreases.
$$f(x)=\left\{\begin{aligned}x+2 . & x<0 \\(x-1)^{2}, & 0<x<3 \\8-x . & 3<x<7 \\2 x-9, & 7<x \\6, & x=0,3.7\end{aligned}\right.$$

Key Concepts

Endpoint Considerations

When dealing with piecewise functions, special attention must be paid to the endpoints or the boundaries of the intervals, especially where the function's definition changes. These points require careful analysis to understand how transitions between different functional expressions affect the overall monotonicity, taking into account the possibility of discontinuities or jumps.

Calculus and Derivative Analysis

In calculus, the first derivative of a function is a fundamental tool for determining monotonicity. The sign of the derivative (positive or negative) indicates whether the function is increasing or decreasing on an interval. Even if a piecewise function is not differentiable at some points, studying its differentiable parts can help determine its local behavior.

Monotonicity

Monotonicity refers to the behavior of a function in terms of its increasing or decreasing nature over an interval. A function is said to be increasing on an interval if, for any two points within the interval, the function's value at the larger point is greater than or equal to its value at the smaller point. Conversely, it is decreasing if the function's value at the larger point is less than or equal to its value at the smaller point.

Piecewise-Defined Functions

A piecewise-defined function is one that is described by different expressions in different parts of its domain. Analyzing such a function requires examining each segment of the domain separately, as the overall behavior can change at the boundaries between pieces. This is particularly important when determining intervals of increase or decrease, as each segment may behave differently.

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