For problems 1-6 do the following: (a) find the critical...

For problems 1-6 do the following: (a) find the critical numbers and critical points; (b) determine the intervals over which $\mathrm{f}$ is increasing and over which $\mathrm{f}$ is decreasing; and (c) identify any relative extrema. $f(x)=\left\{\begin{array}{ll}2 x+9 & \text { if } x \leq-2 \\ x^2+1 & \text { if } x>-2\end{array}\right\}$

For problems 1-6 do the following: (a) find the critical numbers and critical points;
(b) determine the intervals over which $\mathrm{f}$ is increasing and over which $\mathrm{f}$ is decreasing; and (c) identify any relative extrema.
$f(x)=\left\{\begin{array}{ll}2 x+9 & \text { if } x \leq-2 \\ x^2+1 & \text { if } x>-2\end{array}\right\}$

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Key Concepts

Critical Numbers and Critical Points

These are values in the domain of a function where its derivative is zero or undefined, potentially indicating a change in the function's behavior. Critical points are the points on the graph corresponding to these numbers. Identifying these is crucial for understanding where a function might have local maxima or minima.

Increasing and Decreasing Intervals

These intervals are determined by the sign of the derivative. Where the derivative is positive, the function is increasing, and where it is negative, the function is decreasing. Analyzing these intervals helps in understanding the overall behavior of the function and locating relative extrema.

Relative Extrema

Relative, or local, extrema are the points where the function reaches a local maximum or minimum. These are found by examining the critical points and using derivative tests to determine if a critical point is a point of relative maximum or minimum.

Piecewise Functions

A piecewise function is defined by different expressions over different intervals. When analyzing such functions, it is essential to check the behavior at the boundaries where the definition changes, as these points might affect continuity, differentiability, and the existence of critical points.

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