Find all critical values, the largest open intervals on...

Find all critical values, the largest open intervals on which $f$ is increasing, the largest open intervals on which $f$ is decreasing, and all relative maxima and minima. Sketch a rough graph of $f$. In Exercises 37 through 42, assume that the constants $a$ and $b$ are positive. $f(x)=(x+3)^{2 / 3}$

Key Concepts

Critical Points

Critical points are points in the domain of a function where its derivative is either zero or undefined. They are important because they are potential locations where the function can change its direction, signaling possible local maxima, minima, or inflection points.

Chain Rule and Power Rule

For differentiating functions that involve a composition of functions and operations such as raising to a fractional power, the chain rule in combination with the power rule is essential. This provides a systematic way to compute the derivative of complex expressions by differentiating the outer function and then multiplying by the derivative of the inner function.

First Derivative Test

The first derivative test involves examining the sign of the derivative before and after a critical point to determine whether the function is increasing or decreasing in those intervals. This test helps to identify the relative extrema – that is, whether a critical point corresponds to a local maximum or a local minimum.

Relative Extrema

Relative extrema (or local minima and maxima) are specific points in the graph where the function reaches a peak or a trough relative to nearby points. Their identification is based on changes in the sign of the derivative around critical points, as determined by the first derivative test.

Increasing and Decreasing Intervals

Intervals on which a function is increasing or decreasing are determined by analyzing the sign of its derivative. A positive derivative indicates the function is increasing, while a negative derivative indicates the function is decreasing. These intervals help in understanding the overall behavior and shape of the function.

Graphical Analysis

Graphical analysis uses the information about critical points, relative extrema, and intervals of increase or decrease to sketch a rough graph of the function. This visual representation helps in comprehending the function’s overall trend and behavior, providing insights into its continuity and curvature.

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