Question

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

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

Understand the Function

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

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.

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’s no region where it's undefined, it means there’s no relative extrema. From \(y''\), we know that there's a point of inflection at x = 0.

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.