Employment. According to the U.S. Bureau of Labor...

Employment. According to the U.S. Bureau of Labor Statistics, the number of professional services employees Fluctuated during the period $2000-2009,$ as modeled by $E(t)=-28.31 t^{3}+381.86 t^{2}-1162.07 t+16,905.87$ where $t$ is the number of years since $2000(t=0$ corresponds to 2000 ) and $E$ is thousands of employees. Find the relative extrema of this function, and sketch the graph. Interpret the meaning of the relative extrema.

Employment. According to the U.S. Bureau of Labor Statistics, the number of professional services employees Fluctuated during the period $2000-2009,$ as modeled by $E(t)=-28.31 t^{3}+381.86 t^{2}-1162.07 t+16,905.87$
where $t$ is the number of years since $2000(t=0$ corresponds to 2000 ) and $E$ is thousands of employees. Find the relative extrema of this function, and sketch the graph. Interpret the meaning of the relative extrema.

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Key Concepts

Cubic Polynomials

Cubic polynomials are third?degree functions that can display complex behavior, including up to two turning points. They are widely used in modeling because their flexibility allows them to capture fluctuations and trends in data over time.

Critical Points

Critical points of a function are where its derivative is zero or undefined. These points are important because they are potential locations for relative extrema, where the function changes from increasing to decreasing or vice versa.

Relative Extrema

Relative extrema are the local maximum and minimum values of a function. They represent points in the domain where the function reaches a peak or a trough relative to nearby values, which is essential for interpreting trends in practical models.

Derivative Test

The derivative test involves taking the derivative of a function and using its sign to determine whether a critical point is a relative maximum or minimum. This method is fundamental in calculus for analyzing the behavior and stability of functions.

Graph Sketching

Graph sketching is the process of creating a visual representation of a function based on its critical points, inflection points, and overall trend. It aids in understanding how the function behaves over a given domain, thereby providing insights into the modeled phenomenon.

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