Graphical Analysis Use a graphing utility to graph each...

Graphical Analysis Use a graphing utility to graph
each function. Use the graph to find where the function
is increasing and decreasing, and approximate any
relative maximum or minimum values.
(a) $f(x)=x^{2} e^{-x} \quad$ (b) $g(x)=x 2^{3-x}$

Key Concepts

Critical Points and the First Derivative

Critical points are values in the domain of a function where its derivative is zero or undefined, and they are often associated with changes in the increasing or decreasing behavior of the function. Analyzing the first derivative of a function helps in identifying these points, which in turn aid in approximating relative extrema and understanding the overall graph structure.

Relative Extrema

Relative extrema refer to the local maximum and minimum values of a function within a certain interval. These points occur where the function changes its trend—from increasing to decreasing or vice versa—indicating peaks or valleys. Graphical analysis can be used to approximate these points by observing where the function attains its highest or lowest values locally.

Increasing and Decreasing Functions

A function is said to be increasing on an interval if, as the input value grows, the output value also grows; it is decreasing if an increase in the input yields a decrease in the output. Identifying these intervals involves looking at the slope or steepness of the graph, which reflects how the function behaves across different domains.

Graphical Analysis

Graphical analysis is the process of examining the visual representation of a function to understand its behavior, including overall trends, turning points, and intervals where the function increases or decreases. This approach helps in identifying key features of the function without solely relying on algebraic manipulation or calculus.

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