Minima maxima of a graph with asymptote

I get your concern, the derivative changes its sign at that point, which usualy means that there is a local extreme at that point. But if you look at the graph, it’s pretty clear that it’s an asymptote.
This comes from the definition of the local extreme, the function must be DEFINED in that point in order to have an extreme. Here, the function isn’t defined, because you would divide by zero. Graphicaly, extremes usualy look like the tip of a quadratic function, so that is what you should look for.
 
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