Need Help finding critical points and Inflection points. (graphing)

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June 9th 2011, 11:06 AM
mattyc33

Need Help finding critical points and Inflection points. (graphing)

The question asks for the critical points and inflection points of:

(x^2-1)/(x^3+1)

I've tried to take the derivitave and second derivitave but it just got too complicated. If anyone could help me that would be great.

My quick commands menu doesn't seem to be working but the question is quite simple.

Thanks!
June 9th 2011, 11:36 AM
Quacky

Quote:

Originally Posted by mattyc33

The question asks for the critical points and inflection points of:

(x^2-1)/(x^3+1)

I've tried to take the derivitave and second derivitave but it just got too complicated. If anyone could help me that would be great.

My quick commands menu doesn't seem to be working but the question is quite simple.

Thanks!

I don't see any shorthand, I think we're just going to have to use the quotient rule.

So Let $u=x^2-1$

$\frac{du}{dx}=2x$

Let $v=x^3+1$

$\frac{dv}{dx}=3x^2$

We have:

$\dfrac{dy}{dx}=\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}$

$=\dfrac{(x^3+1)(2x)-(x^2-1)(3x^2)}{(x^3+1)^2}$

$=\dfrac{2x^4+2x-3x^4+3x^2}{x^6+2x^3+1}$

$=\dfrac{-x^4+3x^2+2x}{x^6+2x^3+1}$

$=\dfrac{x(-x^3+3x+2)}{(x^3+1)^2}$

$\dfrac{-x(x-2)(x+1)(x+1)}{((x+1)(x^2-x+1))^2}$

$=\dfrac{-x(x-2)}{(x^2-x+1)^2}$

...which I suppose you could differentiate again, although there's got to be a faster approach? If there is, it's at a level that's beyond me, though.
June 9th 2011, 11:39 AM
waqarhaider

expression can written as

y = x^5 - x^3 + x^2 - 1

then y' = 5x^4 - 3x^2 + 2x

for critical points 5x^4 - 3x^2 + 2x = 0 gives two real roots x = 0 and x = -1 now you can do it
June 9th 2011, 12:00 PM
mattyc33

Quote:

Originally Posted by waqarhaider

expression can written as

y = x^5 - x^3 + x^2 - 1

then y' = 5x^4 - 3x^2 + 2x

for critical points 5x^4 - 3x^2 + 2x = 0 gives two real roots x = 0 and x = -1 now you can do it

Unfortunatly, I have no idea how you got that expression (Thinking).
When I plugged what Quacky said into my calculator I got two x values of x=0 and x=2.

Could anyone please explain how he got this expression (Wondering).
June 9th 2011, 12:21 PM
Quacky

Quote:

Originally Posted by mattyc33

Unfortunatly, I have no idea how you got that expression (Thinking).
When I plugged what Quacky said into my calculator I got two x values of x=0 and x=2.

Could anyone please explain how he got this expression (Wondering).

Editted out a sign error with my original response - and to be honest I'm not entirely sure how he got there either.
June 9th 2011, 01:42 PM
mattyc33

Thanks alot for clearing up the critical points.

I'm pretty terrible at my derivitave calculation though so once again i'm having alot of trouble finding the second derivitave in order to find the inflection points.

As Quacky pointed out the first derivitave is -x(x-2)/(x^2-x+1)^2

If anyone could help me that would be great (Worried).
June 9th 2011, 01:44 PM
Quacky

Quote:

Originally Posted by mattyc33

Thanks alot for clearing up the critical points.

I'm pretty terrible at my derivitave calculation though so once again i'm having alot of trouble finding the second derivitave in order to find the inflection points.

As Quacky pointed out the first derivitave is -x(x-2)/(x^2-x+1)^2

If anyone could help me that would be great (Worried).

Again, I think there must be a shorter method, but what have you tried? Let $u=-x^2+2x$ and $v=(x^2-x+1)^2$
June 9th 2011, 01:49 PM
mattyc33

Yeah, I've tried using the quotient rule, it just gets very messy and I'm 99% sure i've been making mistakes in this mess 0.0
June 9th 2011, 01:52 PM
Quacky

Quote:

Originally Posted by mattyc33

Yeah, I've tried using the quotient rule, it just gets very messy and I'm 99% sure i've been making mistakes in this mess 0.0

Please post some effort. Doing the first one took me long enough!
June 9th 2011, 03:14 PM
mattyc33

Well all I have to say is that there must be an easier way:

(-x^2+2x)'(x^2-x+1)^2 - (x^2+2x)(x^2-x+1)' / (x^2-x+1)^4

(-2x+2)(x^2-x+1)^2 - (x^2+2x)[2(x^2-x+1) x (2x-1)] / (x^2-x+1)^4

(-2x+2)(x^4-2x^3+3x^2-2x+1) - (x^2+2x)(4x-2)(x^2-x+1) / (x^2-x+1)^4

(-2x^5+2x^4-6x^3+4x^2-2x+2x^4-4x^3+6x^2-4x+2) - (4x^5-4x^3-4x) / (x^2-x+1)^4

And from there... I have no idea (Headbang).

I'm not saying to try it haha, I know there must be another way, thanks for the help!
June 9th 2011, 03:43 PM
Quacky

Indeed: The roots are horrid!
http://www.wolframalpha.com/input/?i=Differentiate+-x%28x-2%29%2F%28x^2-x%2B1%29^2
June 9th 2011, 03:51 PM
mattyc33

Quote:

Originally Posted by Quacky

Indeed: The roots are horrid!
http://www.wolframalpha.com/input/?i=Differentiate+-x%28x-2%29%2F%28x^2-x%2B1%29^2

Whew thats great thanks!
June 9th 2011, 06:29 PM
Ackbeet

Try this:

$\frac{x^{2}-1}{x^{3}+1}=\frac{(x-1)(x+1)}{(x+1)(x^{2}-x+1)}=\frac{x-1}{x^{2}-x+1},$

for $x\not=-1.$

I think that might simplify things for you a bit.
June 10th 2011, 06:03 AM
waqarhaider

Sorry for the first reply where I missed '/' sign and solved (x^2-1) (x^3+1)
Here is new reply:
expression can be written as

y = ( x - 1 ) / ( x^2 - x + 1 )

then y' = [ ( x^2 - x + 1 ) 1 - ( x - 1 ) ( 2x - 1 ) ] / ( x^2 - x + 1 )^2

gives y' = ( 2x - x^2 ) / ( x^2 - x + 1 )

again y" = [ ( x^2 - x + 1 )^2 ( 2 - 2x ) - ( 2x - x^2 ) 2( x^2 - x + 1 )( 2x - 1 ) ]/ ( x^2 -x + 1 )^4

gives y" = ( 2x^3 - 6x^2 + 2 ) / ( x^2 - x + 1 )^3 and we find point of inflection
June 10th 2011, 06:11 AM
Ackbeet

Your derivatives look correct to me. So what do you get for critical points and points of inflection?

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