Use a graphing utility to graph each function over the...

use a graphing utility to graph each function over the indicated interval and approximate any local maximum values and local minimum values. Determine where the function is increasing and where it is decreasing. Round answers to two decimal places. $$f(x)=-0.4 x^{3}+0.6 x^{2}+3 x-2 \quad(-4,5)$$

Key Concepts

Cubic Functions

Cubic functions are polynomial functions of degree three that typically exhibit an 'S'-shaped graph with up to two turning points. Their characteristics, including the potential for local extrema and points of inflection, make them important for understanding more complex behaviors in polynomial function analysis.

Graphing Utility

A graphing utility is a digital tool that allows users to visualize the behavior of functions over specified intervals. It assists in identifying key features such as local extrema and intervals of increase or decrease by providing approximate numerical results and visual insight into the function’s behavior.

Local Extrema

Local extrema refer to the points on the graph of a function where it attains local maximum or minimum values. These values indicate where the function changes its increasing or decreasing behavior and are typically found by identifying critical points where the derivative is zero or undefined.

Increasing and Decreasing Intervals

Increasing intervals are portions of the function's domain where the function values consistently rise, while decreasing intervals are those where the function values fall. This behavior is linked directly to the sign of the derivative, with positive derivatives indicating increasing intervals and negative derivatives indicating decreasing intervals.

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