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## Homework Statement

[tex]f(x)={\frac {{x}^{2}-2}{{x}^{2}-1}}[/tex]

List the critical points, extrema, points of inflection, and concavity.

## Homework Equations

[tex]f'(x)={\frac {2x}{{x}^{2}-1}}-{\frac { \left( 2\,{x}^{2}-4 \right) x}{

\left( {x}^{2}-1 \right) ^{2}}}[/tex]

[tex]f''(x)={\frac{2}{x^2-1}-\frac{8x^2}{(x^2-1)^2}+\frac{8(x^2-2)x^2}{(x^2-1)^3}-\frac{2(x^2-2)}{(x^2-1)^2}[/tex]

## The Attempt at a Solution

__Extrema__

Solving for f'(x)=0 and checking for discontinuities, I get critical points of -1, 0, and 1.

I can get that far. I'm unsure of how to make the intervals. Would my lowest interval be [tex]-\infty<x<-1[/tex] or [tex]-1<x<0[/tex]? I understand what's happening on the graph, it's just that discontinuities confuse me. :grumpy:

__Concavity__

Solving for f''(x)=0 and checking for discontinuities, I get points of inflection of -1 and 1 (and two non-real answers which are of no use to me.)

Both of these points are discontinuities. How would I make an interval for this? Shouldn't there also be a point of inflection at (0,2)?

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