Problem 36 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Use the information summarized in the table to sketch the graph of \(f\). \(f(x)=x-3 x^{1 / 3}\) Domain: \((-\infty, \infty)\) Intercepts: \(x\) -intercepts: \(\pm 3 \sqrt{3}, 0\) Asymptotes: None Intervals where \(f\) is \(\nearrow\) and \(\searrow: \nearrow\) on \((-\infty,-1) \cup(1, \infty) ;\) \(\searrow\) on \((-1,1)\) Relative extrema: Rel. max. at \((-1,2) ;\) rel. min. at \((1,-2)\) Concavity: Downward on \((-\infty, 0)\); upward on \((0, \infty)\) Point of inflection: \((0,0)\)

Step-by-Step Solution

Answer

To sketch the graph of \(f(x)=x-3x^{\frac{1}{3}}\), first plot the \(x\)-intercepts at \((-3\sqrt{3}, 0)\), \((3\sqrt{3}, 0)\), and \((0, 0)\). The graph is increasing on \((-\infty, -1) \cup (1, \infty)\) and decreasing on \((-1, 1)\). Mark a relative maximum at \((-1, 2)\) and a relative minimum at \((1, -2)\). The graph is concave downward on \((-\infty, 0)\) and concave upward on \((0, \infty)\), with a point of inflection at \((0, 0)\). Using this information, draw the curve of the function through these points and intervals.

Step 1: 1. Domain

The given domain is \((-\infty, \infty)\). This means that the graph of the function extends infinitely to the left and right.

Step 2: 2. Intercepts

The \(x\)-intercepts are given as \(\pm 3 \sqrt{3}, 0\). These are the points where the graph intersects the \(x\)-axis. So, we have three points: \((-3\sqrt{3}, 0)\), \((3\sqrt{3}, 0)\), and \((0, 0)\).

Step 3: 3. Asymptotes

There are no asymptotes given for the function, meaning that the graph doesn't approach any vertical or horizontal lines as it extends to infinity.

Step 4: 4. Increasing and Decreasing Intervals

The function is increasing on the intervals \((-\infty, -1) \cup (1, \infty)\) and decreasing on the interval \((-1, 1)\). This means that the graph rises as we move from left to right on these intervals, and it falls on the interval \((-1, 1)\).

Step 5: 5. Relative Extrema

There is a relative maximum at \((-1, 2)\) and a relative minimum at \((1, -2)\). These are the points where the graph changes its direction from increasing to decreasing or vice versa.

Step 6: 6. Concavity

The graph is concave downward (shaped like an upside-down bowl) on the interval \((-\infty, 0)\) and concave upward (shaped like a bowl) on the interval \((0, \infty)\). This tells us how the graph bends or curves on these intervals.

Step 7: 7. Point of Inflection

There is a point of inflection at \((0, 0)\). This is the point where the graph changes its concavity from downward to upward. Now that we have all the pieces, we can use this information to sketch the graph of the function: 1. Begin by plotting the \(x\)-intercepts and the point of inflection. 2. Draw the graph increasing (rising) from left to right on the intervals \((-\infty, -1) \cup (1, \infty)\), and decreasing (falling) on the interval \((-1, 1)\). 3. Mark the relative maxima at \((-1, 2)\) and the relative minima at \((1, -2)\). 4. Make the graph concave downward on the interval \((-\infty, 0)\) and concave upward on the interval \((0, \infty)\), making sure that the graph passes through the point of inflection \((0, 0)\). By following these steps, we can create an accurate sketch of the given function.

Key Concepts

Intercepts of a Function

Intercepts are critical points where a graph crosses the axes, serving as crucial initial steps in understanding the behavior of functions. For the function f(x) = x - 3x^{1/3}, the x-intercepts are specifically its crossing points with the x-axis and are found when f(x) = 0. The given intercepts, \(\text{\pm} 3 \sqrt{3}, 0\), tell us that the graph touches the x-axis at three distinct points: \((-3\sqrt{3}, 0)\), \((3\sqrt{3}, 0)\), and \((0, 0)\).

Picturing these intercepts on a coordinate plane provides a scaffold for sketching the entire graph. When we plot these points, they act as guideposts around which the curve of the graph will take shape. While the y-intercept, in this case, is also at \((0, 0)\), it is usually found by setting x = 0 in the function and solving for f(x). Intercepts provide a snapshot of where the function's graph will touch or cross the coordinate axes, which is an integral part of the visualization when sketching graphs.

Concavity and Inflection Points

Concavity reveals the direction in which a graph curves, and inflection points indicate where it changes curvature. For our function f(x), it is concave downwards on \((-fty, 0)\) and concave upwards on \((0, fty)\), resembling an upside-down bowl when downwards and a right-side-up bowl when upwards. This aspect of graphing is critical because it gives us insight into the nature of the graph's curvature.

Concavity is typically determined by the second derivative of the function, if available. Another method, without relying on calculus, involves looking at the increasing or decreasing nature of the segments and inferring concavity. Inflection points occur at values where concavity changes, which in this scenario is at \((0, 0)\). This is not merely a point where the graph crosses itself, but where it undergoes a fundamental change in behavior—from bowing out to bowing in, or vice versa. Recognizing and graphically representing concavity and inflection points is essential to sketch an accurate graph.

Relative Extrema of a Function

Relative extrema are the highest or lowest points in a particular segment of a graph, also considered as local maxima or minima. They represent the peaks or troughs where a function turns around. In our exercise, a relative maximum is located at \((-1, 2)\) and a relative minimum at \((1, -2)\).

The presence of a relative maximum means the function increases until that point and then begins to decrease, while a relative minimum indicates the function decreases to that point before increasing. These points occur at the top or bottom of 'hills' and 'valleys' on the graph of a function. Plotting these points is crucial as they provide a clear indication of how the graph's shape changes direction. Relative extrema are typically discovered by analyzing the first derivative of a function, but they can also be inferred from the given information about where the function is increasing and decreasing. Including these in your graph gives a more precise depiction of the function's behavior in different intervals.