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Chapter 9: Problem 27

Use a graphing utility to graph the function. Choose a window that allows all relative extrema and points of inflection to be identified on the graph. \(y=1-x^{2 / 3}\)

Short Answer

Expert verified
The graph of the function \(y=1-x^{2 / 3}\) is U-shaped with an opening downwards. The global and local minimum is at the vertex of the curve. There are no points of inflection, as the curve does not change concavity.

Step by step solution

01

Understanding the Function

First, it's important to understand what kind of function you have here: \(y=1-x^{2 / 3}\). This is a power function with a negative sign in front of the variable term. Overall, the graph is expected to be U-shaped with the opening downwards because of the negative sign.
02

Using the Graphing Utility

Next, input the function \(y=1-x^{2 / 3}\) into the graphing utility. Depending on the particular utility, it's usually as simple as typing in the expression '1-x^(2/3)'.
03

Setting the Window

Now, adjust the window to view all the necessary points. Due to the fractional exponent, the graph is mostly defined for all real numbers, so set the window to include negative and positive values of 'x' as well as 'y'.
04

Identifying Relative Extrema and Inflection Points

Once the graph is displayed, look for any relative extrema 鈥� these are points on the graph where the function reaches a local minimum or maximum. In this case, the global minimum point also serves as a local minimum, which is the vertex of the u-shaped curve. For this function, there are no inflection points as the curve does not change concavity.

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Most popular questions from this chapter

The table lists the average monthly Social Security benefits \(B\) (in dollars) for retired workers aged 62 and over from 1998 through 2005 . A model for the data is \(B=\frac{582.6+38.38 t}{1+0.025 t-0.0009 t^{2}}, \quad 8 \leq t \leq 15\) where \(t=8\) corresponds to 1998 . $$ \begin{array}{|l|l|l|l|l|l|l|l|l|} \hline t & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 \\ \hline B & 780 & 804 & 844 & 874 & 895 & 922 & 955 & 1002 \\ \hline \end{array} $$ (a) Use a graphing utility to create a scatter plot of the data and graph the model in the same viewing window. How well does the model fit the data? (b) Use the model to predict the average monthly benefit in \(2008 .\) (c) Should this model be used to predict the average monthly Social Security benefits in future years? Why or why not?

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