just wondering if anyone knows how to determine the inflection points,max/min and concavity of the rational function:
f(x)=x2+4/x2-9
just finished finding the vertical and horizontal asymptotes, symmetry, x and y intercepts, and domain, but can't recall the meathods for determining the above. I used to know how to do this like the back of my hand!
You do it just like you do it for any other function.Originally Posted by dcfi6052
just wondering if anyone knows how to determine the inflection points,max/min and concavity of the rational function:
f(x)=x2+4/x2-9
just finished finding the vertical and horizontal asymptotes, symmetry, x and y intercepts, and domain, but can't recall the meathods for determining the above. I used to know how to do this like the back of my hand!
Relative max/min:
Set f'(x) = 0. The values of x where this happens are your candidates.
So f'(x) = 0 only when x = 0.
Now is this a relative max or a relative min? You can find out by graphing, but we need the second derivative to get the other stuff anyway, so let's do the second derivative test on this.
(after a bit of factoring)
So what is f''(x) for x = 0?. Since f''(0) is negative, the point (0, -4/9) is a relative maximum.
As it happens, the second derivative IS the concavity, so you have already found this.
An inflection point is where f''(x) = 0. So this happens when, which has no real solutions for x. Thus there are no inflection points.
-Dan
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