How do I know for sure this function has no absolute max/min without drawing the graph?

[itsrayex]

After doing all of this I thought 'well, here's the absolute min' but then I checked with a function plotter and the function looks like this, meaning there is no absolute min/max:


Is there a way to tell by just looking at the little scheme I drew on the right, without going through all the steps to draw a graph?
Before seeing the graph I knew that this is a function of the kind x^-2 and is therefore not continuous. How does this help?

 

[Dr.Peterson]

View attachment 31084
After doing all of this I thought 'well, here's the absolute min' but then I checked with a function plotter and the function looks like this, meaning there is no absolute min/max:
View attachment 31085

Is there a way to tell by just looking at the little scheme I drew on the right, without going through all the steps to draw a graph?
Before seeing the graph I knew that this is a function of the kind x^-2 and is therefore not continuous. How does this help?

You can see almost at a glance that it has two vertical asymptotes (at 1 and -1), both with odd multiplicity, which implies it will go to both positive and negative infinity.

Voila! No absolute max or min.

And this would show up on your "scheme" if you included information about places where the function or its derivative is undefined, as I've recommended.

 

[blamocur]

Show that for any positive [imath]M[/imath] one can find [imath]x_1[/imath] such that [imath]f(x_1) > M[/imath] and [imath]x_2[/imath] such that [imath]f(x_2) < -M[/imath]