I need to find a maximum/minimum value of a function, and my basic calculus has failed me.
I have solved for an eiganvalue in terms of $a$, and need to know when that magnitude of that eigenvalue falls within a certain range based on $a$.
$$-4 \leq |\pm \sqrt{a^2-2a-1}-a |\leq 0$$
In other words, I also have to consider the complex part of this inequality and I am completely stumped.
I tried taking the limit of the thing to see if there was an assymptote but I can't even figure out how to do that. I've tried taking the derivative of the thing to see if it has a max or min that falls inside or outside the inequality. I've exhausted all the tricks I remember from calc 2. My professor doesn't like to help on these things, and I've spent 3-4 hours on this single part of a single question on a four part homework assignment. He would just tell me to "think more deeply about it", and then when I can't figure it out in time to turn it in, I'll lose all points even if I have work shown. I'm exhausted. Sorry, end rant.