MHB Solve Image of f(x)=-2(x+1)/(x-1)^2 in x from [0,1)U(1,3]

[Vali]

I have the next function:
f(x)= -2(x+1)/(x-1)^2
x is from [0,1)U(1,3]
I need to find the image of the function which is (-infinity, -2].I tried with the derivative but when I solve f'(x)=0 I obtain x=-3 which is not from x interval.I don't know how to continue.

 

[Olinguito]

I presume you got
$$f'(x)\ =\ \frac{2(x+3)}{(x-1)^3}$$
which is correct. Notice that $f'(x)<0$ when $x\in[0,\,1)$ and $f'(x)>0$ when $x\in(1,\,3]$. So $f(x)$ is decreasing on $[0,\,1)$ and increasing on $(1,\,3]$. Also, $f(0)=f(3)=-2$ and $f(x)\to-\infty$ as $x$ tends to $1$ from either side. From these, you should be able to piece together the range of $f(x)$.

 

[Vali]

Thanks a lot!