Question

In Exercises \(75-86\), use a computer algebra system to analyze the graph of the function. Label any extrema and/or asymptotes that exist. $$ f(x)=\frac{x}{x^{2}-4} $$

Short answer

After performing the steps, you should be able to identify that there are no extrema, vertical asymptotes are at x = -2 and x = 2, and the horizontal asymptote is at y = 0.

Step-by-step solution

Differentiate the function

Find the derivative of \(f(x) = \frac{x}{x^{2}-4}\) using the quotient rule. Set the derivative equal to zero and solve for x to find the stationary points, which will help identify the extrema.

Find the critical points and analyze them

Plug these x-values from step 1 into the original function to get their corresponding y-values. Check the points to the left and right of these stationary points in the derivative to find out whether these points are maxima or minima.

Find the asymptotes

Vertical asymptotes can be found by setting the denominator of the function equal to zero and solving for x. Horizontal asymptotes can be found by finding the limit of the function as x tends to plus and minus infinity.