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Chapter 9: Problem 29

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. \(y=x^{1 / 3}+1\)

Short Answer

Expert verified
The function \(y=x^{1/3}+1\) has no extrema or inflection points. The graph is a smooth, rising curve that passes through the point (0,1) and extends from negative to positive infinity in both the x and y directions.

Step by step solution

01

Input Function

To graph the function, first input the function \(y=x^{1/3}+1\) into a graphing utility.
02

Choose Appropriate Window

Next, choose a window that allows all relative extrema and inflection points to be shown. Since the function given is a cube root function, the graph of the function will extend from negative to positive infinity in both the x and y directions. So, a wide window is recommended, such as x: [-10, 10] and y: [-10, 10].
03

Identify Extrema and Inflection Points

The function \(y=x^{1/3}+1\) has neither relative extrema nor inflection points. The graph has a smooth curve but it does not turn, meaning no extrema (minimum or maximum points) or points of inflection (points where the curve changes concavity) are present. Hence, the final graph should resemble a smoothly increasing curve that emerges from the negative x-axis, passes through the points (0,1) and continues to rise along the positive x-axis. Note that the '1' in the given function shifts the graph upward by one unit.

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Most popular questions from this chapter

Marginal Analysis, use differentials to approximate the change in cost, revenue, or profit corresponding to an increase in sales of one unit. For instance, in Exercise 29, approximate the change in cost as \(x\) increases from 12 units to 13 units. Then use a graphing utility to graph the function, and use the trace feature to verify your result. \(P=-x^{2}+60 x-100 \quad x=25\)

Use a graphing utility to graph the function. Explain why there is no vertical asymptote when a superficial examination of the function may indicate that there should be one. \(g(x)=\frac{x^{2}+x-2}{x-1}\)

Compare the values of \(d y\) and \(\Delta y\). \(y=2 x+1 \quad x=2 \quad \Delta x=d x=0.01\)

The cost function for a certain model of personal digital assistant (PDA) is given by \(C=13.50 x+45,750\), where \(C\) is measured in dollars and \(x\) is the number of PDAs produced. (a) Find the average cost function \(\bar{C}\). (b) Find \(\bar{C}\) when \(x=100\) and \(x=1000\). (c) Determine the limit of the average cost function as \(x\) approaches infinity. Interpret the limit in the context of the problem.

Sketch the graph of the function. Label the intercepts, relative extrema, points of inflection, and asymptotes. Then state the domain of the function. \(y=\frac{x^{4}}{x^{4}-1}\)
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