Let [math], for [math].

Using a calculator or computer, sketch the graph of [math] for [math]. Describe what happens as [math] changes.

[math] has a local minimum. Find the location of the minimum.
[math]

Find the [math]-coordinate of the minimum.
[math]

Find the value of [math] for which this [math]-coordinate is largest.
[math]

How do you know that this value of [math] maximizes the [math]-coordinate? Find [math] to use the second-derivative test.
[math]
(Note that the derivative you get is negative for all positive values of [math], and confirm that you agree that this means that your value of [math] maximizes the [math]-coordinate of the minimum.)