curve sketching: f(x) = x / (x - 1)^2

[Guest]

f(x) = x / (x -1 )^2

DOMAIN
x cannot equal 1

INTERCEPTS
x-int
0=x/(x-1)^2
x=0

y-int
y=0/(0-1)^2
y=0

SYMMETRY
f(-x)=-x/(-x-1)^2
-no symmetry

-f(-x)=-(x/(x-1)^2
=-

ASYMPTOTES

...then im supposed to find the max/min values and the concavity which im pretty sure i did correctly however im not sure if i did the stuff up there correctly. Thanks for any help.

 

[soroban]

Hello, bandaid-bandet!

\(\displaystyle f(x) \:= \:\frac{x}{(x -1 )^2}\)

DOMAIN: \(\displaystyle \:x\,\neq\,1\)

INTERCEPTS
x-int: \(\displaystyle \:0\:=\:\frac{x}{(x-1)^2}\;\;\Rightarrow\;\;x\,=\,0\)

y-int: \(\displaystyle \:y\:=\:\frac{0}{(0-1)^2}\;\;y\,=\,0\)

SYMMETRY
\(\displaystyle f(-x)\:=\:\frac{-x}{(-x-1)^2}\) . . . no symmetry

\(\displaystyle \,-f(-x)\:=\:-\frac{x}{(x-1)^2}\) . . . no symmetry

Then I'm supposed to find the max/min values and the concavity
. . which I'm pretty sure I did correctly.
However, I'm not sure if I did the stuff up there correctly. . You did great!

ASYMPTOTES

You already found the vertical asymptote: \(\displaystyle \,x\,=\,1\)

Horizontal asymptote: \(\displaystyle \L\:\lim_{x\to\infty}y\;=\;\lim_{x\to\infty}\frac{x}{(x\,-\,1)^2}\:=\:0\)
. . Hence: \(\displaystyle \:y\,=\,0\) (x-axis)


MAX\MIN
We have: \(\displaystyle \:f'(x)\:=\:-\frac{x\,+\,1}{(x\,-\,1)^3}\;\;\Rightarrow\;\;x\,=\,-1\) . . . critical value

And we have: \(\displaystyle \:f''(x)\:=\:\frac{2(x\,+\,2)}{(x\,-\,1)^4}\)

At \(\displaystyle x\,=\,-1:\;\;f''(-1)\:=\:\frac{2(-1\,+\,2)}{(-1\,-\,1)^4}\:=\:+\frac{1}{8}\) . . . positive, concave up: \(\displaystyle \,\cup\)

. . Hence, a minimum at \(\displaystyle \left(-1,\,-\frac{1}{4}\right)\)


CONCAVITY
We have: \(\displaystyle \:f''(x)\:=\:\frac{2(x\,+\,2)}{(x\,-\,1)^4}\)

Inflection point at \(\displaystyle x\,=\,-2,\;y\,=\,-\frac{2}{9}\)

When \(\displaystyle x\,<\,-2:\;\;f''(x) \,<\,0\) . . . concave down
When \(\displaystyle x\,>\,2:\;\;f''(x)\,>\,0\) . . . concave up


The graph looks like this:

                                |           :
                                |          *:*
                                |           :
                                |         * : *
                                |        *  :  *
                                |      *    :     *
           -2   -1              |   *       :           *
    --------+----+--------------*-----------+-------------- 
     *                      *   |           :1
            *          *        |           :
                 *              |           :
                                |           :