91Ó°ÊÓ

In Exercises \(1-4\), use the information summarized in the table to sketch the graph of \(\bar{f}\). $$ f(x)=x-3 x^{1 / 3} $$ $$ \begin{array}{|l|l|} \hline \text { Domain } & (-\infty, \infty) \\ \text { Intercepts } & x \text { -intercepts: } \pm 3 \sqrt{3}, 0 ; \\ & y \text { -intercept: } 0 \\ \text { Symmetry } & \text { With respect to the origin } \\ \text { Asymptotes } & \text { None } \\ \text { Intervals where } f \text { is / or } \backslash & \begin{array}{l} \prime \text { on }(-\infty,-1) \text { and on } \\ (1, \infty) ; \backslash \text { on }(-1,1) \end{array} \\ \text { Relative extrema } & \begin{array}{l} \text { Rel. max. at }(-1,2) ; \\ \text { rel. min. at }(1,-2) \end{array} \\ \text { Concavity } & \begin{array}{l} \text { Downward on }(-\infty, 0) \\ \text { upward on }(0, \infty) \\ \text { Point of inflection } & (0,0) \end{array} \\ \hline \end{array} $$

Step by step solution

01

1. Plot the Intercepts

First, plot the x-intercepts and y-intercept on the coordinate plane: x-intercepts: \(\pm 3\sqrt{3}\), \(0\) y-intercept: \(0\)

02

2. Symmetric with respect to the origin

The table tells us that the function \(f(x)\) is symmetric with respect to the origin. This means if we have a point \((x, y)\) on the graph, then we also have the point \((-x, -y)\). This information will help us in plotting the graph more accurately.

03

3. Increasing and Decreasing

According to the table, \(f(x)\) is increasing on the intervals \((-\infty, -1)\) and \((1, \infty)\), and decreasing on the interval \((-1, 1)\).

04

4. Relative Extrema

We have been given two relative extremum points in the table: - Relative maximum at \((-1, 2)\) - Relative minimum at \((1, -2)\) Plot these points on the graph.

05

5. Concavity and Inflection Point

The table provides information about the concavity of the graph: - Concave downward on \((-\infty, 0)\) - Concave upward on \((0, \infty)\) There should be a point of inflection at \((0, 0)\), where it changes from concave downward to concave upward.

06

6. Sketch the Graph

Using all of the above information and plotted points, sketch a smooth curve representing the function. Make sure to maintain the symmetry, increasing and decreasing intervals, relative extremum points, and concavity information given in the table.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Function Symmetry

Understanding function symmetry is crucial for sketching graphs accurately. In our exercise, the function f(x) = x - 3x^{1/3} is symmetric with respect to the origin. This type of symmetry, known as origin symmetry or odd symmetry, means that if a point (x, y) belongs to the graph, then the point (-x, -y) also belongs to it.

When graphing, this information can assist you in predicting the shape of the other half once one side is plotted. If a function is symmetrical about the origin, its graph will intersect the origin and display mirror-like qualities across both the x-axis and y-axis.

Increasing and Decreasing Intervals

Identifying intervals where the function is increasing or decreasing adds depth to our understanding of the graph's shape. The function increases where its slope is positive, and decreases where its slope is negative. For the given function, it increases on both (-fty, -1) and (1, fty), and decreases on (-1, 1).

This tells us that the function rises as we move from left to right in the intervals of increase and falls in the interval of decrease. Graphing these intervals correctly ensures the graph will have the correct trend at every point.

Relative Extrema

Relative extrema refer to the high and low points on the function within a particular interval around them, known as local maximums and minimums, respectively. In our exercise, there's a relative maximum at (-1, 2) and a relative minimum at (1, -2).

This tells you that at x = -1, the function peaks before starting to decrease, and at x = 1, the function reaches a trough before beginning to rise. Plotting these points provides anchor spots around which the rest of the graph is shaped.

Concavity and Inflection Points

Concavity of a graph is about the direction the curve opens. If the curve opens upwards like a cup, it is concave upward, and if it opens downwards like a frown, it is concave downward. The function f(x) in this exercise is concave downward on (-fty, 0) and concave upward on (0, fty). An inflection point occurs where the graph changes concavity, in this case at (0, 0).

An inflection point is not always a point where the function has a relative extremum, rather it's a point where the curvature changes. The curve changes from showing a frown before x=0 to a smile after x=0. This crucial characteristic helps to delineate the shape of the function's graph accurately.