Question

Find any relative extrema of the function. Use a graphing utility to confirm your result. \(g(x)=x \operatorname{sech} x\)

Short answer

Without computing tools it's not possible to identify the exact value for \(x\) in step 2 of the solution, thus it is recommended to use a graphing utility to identify the relative extrema.

Step-by-step solution

Compute the derivative

Differentiating \(g(x)=x \operatorname{sech} x\) using the product rule, we have \(g'(x)= \operatorname{sech} x - x \operatorname{sech} x \operatorname{tanh} x\).

Set the derivative equal to zero

Find the points where \(\operatorname{sech} x - x \operatorname{sech} x \operatorname{tanh} x = 0\). Then solve for \(x\). The solutions to this equation are the candidates for the relative extrema.

Examining the number line

Once we have the \(x\)-values from step 2, we examine the number line and determine whether the derivative changes sign at these points. If there is a change in the sign of the derivative, we have a relative extremum at the point.

Confirm the results

After we identify all potential extrema, we should cross-check using a graphing utility. This will confirm the results obtained in previous steps.