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Chapter 3: Problem 84

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) \(f(x)=x^{2} e^{-x}\) (b) \(g(x)=x 2^{3-x}\)

Short Answer

Expert verified
From the graph plots, \(f(x)=x^{2} e^{-x}\) is increasing until \(x=0\) and then decreasing from \(x=0\) to \(+\infty\). \(g(x)=x 2^{3-x}\) is also increasing until \(x=0\) and then decreasing for \(x>0\). Both functions have their relative maximum at \(x=0\).

Step by step solution

01

Function Analysis: f(x)

Using a graphing utility, start by plotting the function \(f(x)=x^{2} e^{-x}\). Carefully observe the plot, and note that the plot curve increases until it reaches a maximum point and then decreases. Hence the function is increasing from (-鈭�,0) until it reaches a peak. Beyond this maximum point, the function starts to decrease towards (-鈭�,+鈭�). The maximum point identified from our plot is the relative maximum.
02

Function Analysis: g(x)

Similar to step one, we now plot the function \(g(x)=x 2^{3-x}\). As with the previous step, examine the curve. Notice how it first increases up to a point before decreasing. This function is increasing from (-鈭�,0) and then starts decreasing from (0,+鈭�). A maximum point can be identified from the plot, and this relative maximum is the peak point of the graph.

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