Question

How do you find the inflection points for the function `f(x)=x/(x-1)`?

Answer 1

Unfortunately, this rational function does not have any vertical asymptote.

Remember that an inflection point is a point of a curve where its concavity changes.

By Quotient Rule,
`f'(x)={1cdot(x-1)-x cdot1}/{(x-1)^2}={-1}/{(x-1)^2}=-(x-1)^{-2}`
By General Power Rule,
`f''(x)=2(x-1)^{-3}=2/{(x-1)^3}`
Since `f''(x)<0` when `x<1` and `f''(x)>0` when `x>1`, there is a concavity change only at `x=1`; however, the original function `f(x)` is undefined at `x=1`, so it cannot have an inflection point there. Hence, there is no inflection point.