Find: (a) the intervals on which f is increasing, (b) the...

Find: (a) the intervals on which f is increasing, (b) the intervals on which f is decreasing, (c) the open intervals on which f is concave up, (d) the open intervals on which f is concave down, and (e) the x-coordinates of all inflection points.
$f(x)=x^{2 / 3}-x$

Key Concepts

Inflection Points

Inflection points are points on the graph of a function where the concavity changes from up to down or vice versa. Identifying these points involves solving for where the second derivative is zero or undefined and confirming that there is a sign change, indicating a change in the curvature of the graph.

Concavity

Concavity describes the direction in which a function's graph curves. It is determined by the second derivative: if the second derivative is positive, the function is concave up (curve opens upward), and if it is negative, the function is concave down (curve opens downward). This analysis helps in understanding the curvature of the function and predicting the behavior between critical points.

Derivative Tests

The derivative tests involve employing the first derivative test to determine intervals of increase and decrease, and the second derivative test to assess concavity and identify inflection points. These tests are essential methods in calculus for analyzing the behavior of functions without necessarily plotting the graph.

Monotonicity

Monotonicity refers to the increasing or decreasing behavior of a function over an interval. By analyzing the first derivative of the function, one can determine where the function is increasing (first derivative positive) or decreasing (first derivative negative), which is essential for understanding its overall behavior and locating potential local extrema.

Critical Points

Critical points are values in the domain of a function where the first derivative is zero or undefined. They are important in determining the behavior of the function because they often indicate potential locations of local minima, local maxima, or points of inflection where the behavior of the function changes.

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