Hi all,
I hate to ask this, but my puny brain has been stumped on this for a while.
I'm given the function:
[ (x^2) - 4) ] / [ (x^2) - 9) ], and am asked to determine the intervals of concavity.
I had no issue figuring out the first and second derivatives, seeing as the answers confirmed them to be correct:
f'(x) = (-10x) / [ (x^2) - 9 ]^2
f''(x) = 30[ (x^2) + 3) ] / [ (x^2) - 9) ]^3
However, I come across an issue when trying to figure out the intervals where the function is concave up or concave down algebraically. I usually set f''(x) equal to zero (or f''(x) <0, f''(x) > 0) to figure this out, but I haven't been able to do this here since there doesn't seem to be a solution. Is there another way to figure out the intervals of concavity here without having to sketch the graph?
TOO LONG TO READ: how do I solve f''(x) = 0 (f''(x) given above) or figure out concavity given f''(X) above?
Really would appreciate any help here, thank you!