Question

In Exercises \(25-32,\) use a computer algebra system to find the derivative of the function. Then use the utility to graph the function and its derivative on the same set of coordinate axes. Describe the behavior of the function that corresponds to any zeros of the graph of the derivative. \(y=\frac{\sqrt{x}+1}{x^{2}+1}\)

Short answer

The derivative of the function \(y=\frac{\sqrt{x}+1}{x^{2}+1}\) is \( y'=\frac{x(1-\sqrt{x})}{(x^{2}+1)^{2}}\). Plotting this derivative graph, the points where the derivative intersects the x-axis indicate where the original function changes concavity (from increasing to decreasing or vice versa) as represented by horizontal tangents at these points.

Step-by-step solution

Compute the Derivative of the Function

Utilize a computer algebra system (CAS) to compute the derivative of the function \(y=\frac{\sqrt{x}+1}{x^{2}+1}\). This can usually be done by inputting the function into the CAS and selecting the command or operation to differentiate. An example of the derivative would be \( y'=\frac{x(1-\sqrt{x})}{(x^{2}+1)^{2}}\).

Graph the Function and Its Derivative

Use the computer algebra system to graph the original function and its derivative on the same set of coordinate axes. Observe the graphs of both function and derivative, watching out for points where the derivative graph intersects the x-axis (indicating a derivative value of zero).

Analyze the Zeros of the Derivative

The zeros of the derivative represent points where the original function has horizontal tangents (i.e., 'peaks' or 'troughs'). At these points, the original function typically switches from increasing to decreasing or vice versa, since the slope of the tangent line (denoted by the derivative value) changes sign.