Find any extrema, points of inflection, asymptotes, and symmetry for function.
f(x) = (x^5-10x^3+9x) / ( x^4 - 16)
The Attempt at a Solution
Extrema: I took the first derivative by using the Quotient Rule, and got
(x^8 + 10x^6 - 107x^4 + 480x^2 - 144) / ( x^4 - 16)^2
I know that to find an extrema, I need to determine the critical numbers. Which are when f ' is equal to 0 or is undefined. I determined that "2" makes f ' undefined, but it also is not defined in the original function, f(x), so that is not a critical number. But I cannot for the life of me figure out how to factor the numerator when set to 0.
I tried to graph f ' , and it seems like x = 0 is a critical number, but when i plug it into the numerator it gives me -144...I feel like I am missing something, can someone please help me figure out how to determine the critical numbers please? I think I'm having more algebra issues than calculus.
Then there's the possibility that I took the wrong first derivative. If someone could check me on that, I would be thankful.