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Chapter 9: Problem 24

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. \(y=\frac{x}{x^{2}+1}\)

Short Answer

Expert verified
The function \(y=\frac{x}{x^{2}+1}\) does not have any local extrema. There is a point of inflection at (0, 0).

Step by step solution

01

Understand the Function

We are given the function \(y=\frac{x}{x^{2}+1}\). This is a rational function where x is divided by \(x^{2}+1\).
02

Locate the Extrema and Inflection Points

To locate the extrema and inflection points of a function, first need to find the first and second derivatives of the function. In this case, the first derivative of \(y=\frac{x}{x^{2}+1}\) is \(y'=\frac{1}{(x^{2} + 1)^{2}}\). The second derivative is \(y''=-\frac{2x}{(x^{2} + 1)^{3}}\). Critical points occur when \(y'\) is zero or undefined, and the points of inflection when \(y''\) is zero or undefined. Solving these equations: for y', there's no real root. For y'', the root is x = 0.
03

Identify the Extrema and Inflection Points

By plugging the root x = 0 into the derivative formulas, the extrema and inflection points can be identified. Here, since the derivative \(y'\) never equals zero and there鈥檚 no region where it's undefined, it means there鈥檚 no relative extrema. From \(y''\), we know that there's a point of inflection at x = 0.
04

Graph the function

Now utilize graphing software to plot the function \(y=\frac{x}{x^{2}+1}\) in a window that allows all relative extrema and points of inflection, which is only at the point (0,0), to be seen. The graph will show the standard form for this equation. Make sure to verify that inflection point on the graph.

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Most popular questions from this chapter

The cost \(C\) (in dollars) of producing \(x\) units of a product is \(C=1.35 x+4570\) (a) Find the average cost function \(\bar{C}\). (b) Find \(\bar{C}\) when \(x=100\) and when \(x=1000\). (c) What is the limit of \(\bar{C}\) as \(x\) approaches infinity?

Find the differential \(d y\). \(y=(4 x-1)^{3}\)

A manufacturer determines that the demand \(x\) for a product is inversely proportional to the square of the price \(p\). When the price is \(\$ 10\), the demand is 2500\. Find the revenue \(R\) as a function of \(x\) and approximate the change in revenue for a one-unit increase in sales when \(x=3000\). Make a sketch showing \(d R\) and \(\Delta R\).

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. \(C=0.05 x^{2}+4 x+10 \quad x=12\)

The side of a square is measured to be 12 inches, with a possible error of \(\frac{1}{64}\) inch. Use differentials to approximate the possible error and the relative error in computing the area of the square.
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