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Chapter 5: Problem 78

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
Graphing functions and analyzing their behavior helps to determine where they are increasing and decreasing, and to find maximum or minimum values. Function \(f(x)=x^{2} e^{-x}\) and function \(g(x)=x 2^{3-x}\) both have distinct graphs that indicate these features, and using a graphing utility facilitates this identification process.

Step by step solution

01

- Graph Function (a)

Use a graphing utility to input the function \(f(x)=x^{2} e^{-x}\). Observe and sketch the resulting output as accurately as possible. Observe the shape of the graph, the x-intercepts and overall direction of the function.
02

- Analyze Function (a)

Analyze the graph. The function is increasing when the line moves upwards as you go from left to right and decreasing when it moves downwards. Relative maximum values are the highest points in a certain part of the graph, and relative minimum values are the lowest points. Locate these on the graph.
03

- Graph Function (b)

Repeat the process for function \(g(x)=x 2^{3-x}\). Input this function into the graphing utility and sketch the output. Look at the shape, x-intercepts, and overall direction.
04

- Analyze Function (b)

Again, identify where the function is increasing and decreasing, and locate any relative maxima or minima. The function is increasing when the line moves upwards from left to right and decreasing when it moves downwards. Maxima and minima should be looked at in a similar manner as was done for function (a)

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Most popular questions from this chapter

\(f(x)=3 \ln x\) \(x=0.32\)

In Exercises 101 and 102, (a) use a graphing utility to graph the function, (b) use the graph to determine the intervals in which the function is increasing and decreasing, and (c) approximate any relative maximum or minimum values of the function. \(f(x)=|\ln x|\)

\(\ln 679=6.520 \ldots\)

\(e^{2 x}=3\)

\(81^{1 / 4}=3\)
See all solutions

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