Extrema and concavity of a function.

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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)?
 
Last edited:

Answers and Replies

  • #2
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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:

The function seems to be perfectly well defined when x<-1.


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.)

Are you sure? That function sure looks like it could change it's concavity.

How do you make intervals? How about [tex] -\infty < x < -1[/tex], [tex] -1 < x < 1 [/tex] and [tex] 1 < x < \infty [/tex]? What's wrong with them? :)
 
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The function seems to be perfectly well defined when x<-1.




Are you sure? That function sure looks like it could change it's concavity.

How do you make intervals? How about [tex] -\infty < x < -1[/tex], [tex] -1 < x < 1 [/tex] and [tex] 1 < x < \infty [/tex]? What's wrong with them? :)

The extrema makes sense, thank you for that. I guess my brain was fried last night. :redface:

The [tex]-1 < x < 1[/tex] interval for the concavity still don't make sense to me, though. If I used a test value of x=0 for said interval, wouldn't that give me the concavity for the parabola on the upper part of the graph? So shouldn't there be another point of concavity at (0,2)? But there obviously isn't, since f''(0)=2 (ie, it's not a point of inflection.) I guess that's what is confusing me.

http://img42.imageshack.us/img42/5492/grapho.jpg [Broken]
 
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  • #4
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For a parabola f''(x) = constant for all x. However you are right in that there's something going on in the interval (-1,1). Perhaps you should check your calculations again.

For motivation, check for example f''(0.5) and f''(0.7).
 
  • #5
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For a parabola f''(x) = constant for all x. However you are right in that there's something going on in the interval (-1,1). Perhaps you should check your calculations again.

For motivation, check for example f''(0.5) and f''(0.7).

edit:

Ok it just clicked in my head. There's a point of inflection whenever f''=0 or DNE. At (0,2) f''[tex]\neq[/tex]0, so there isn't a point of inflection there. Because the parabola is bowl-shaped I was thinking there had to be a point of concavity there, but there isn't.
 
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