Question

Find any relative extrema and inflection points of the function. Use a graphing utility to confirm your results y= x ln x

Answer 1

First: because of the ln-function, x > 0.
Differentiate:\[\frac{ dy }{ dx }=1*\ln x+x* \frac{ 1 }{ x }=\ln x+1\]
Now solve\[\frac{ dy }{ dx }=0\]to find the x-value(s) where the graph has a horizontal tangent line.
With the aid of a sign scheme of the derivative you can establish whether these x-values are related to a min/max value or that there is an inflection point.

To find additional inflection points, you should differentiate once more, and solve y''=0.
You get an inflection point if there is a sign change around the solution of y''=0.