I am wondering if I am starting the curve sketching right

frosty8688

1. Sketch the curve



2. [itex]\frac{x}{(x-1)^{2}}[/itex]



3. Here's what I have; the domain is all x except x ≠ 1, y-intercept of 0, x-intercept of 0, symmetric about the origin since it is an odd function, horizontal asymptote of 0, vertical asymptote of 1, and for the increasing and decreasing of the function, I'm wondering if I did it right. f'(x) = [itex]\frac{1*(x-1)^{2}-x*2(x-1)*1}{(x-1)^{4}}[/itex] = [itex]\frac{(x-1)^{2}-2x(x-1)}{(x-1)^{4}}[/itex] = after factoring out the one (x-1) [itex]\frac{-x-1}{(x-1)^{3}}[/itex] Increasing on (-1,1) and decreasing on (-∞,-1), (1,∞). There is no local maximum, because f(1) would result in division by 0 which is impossible. Local minimum of f(-1) = [itex]\frac{-1}{4}[/itex]

Dick

A lot of that is right. But it's certainly not an odd function and symmetric around the origin, is it? Some of the other stuff you've said would contradict that.

frosty8688

Based on using f(-x) would result in it being negative divided by positive which would make it an odd function, because it would be f(-x) = -f(x).

Dick

f(1) is undefined. f(-1)=(-1/4) (as you said). That doesn't sound symmetric or odd to me.
(x-1)^2 is not an even function.

frosty8688

So the answer would be neither odd or even and no symmetry.

Dick

Did you sketch the graph? It doesn't look even or odd or symmetric to me. If you agree, then yes.