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

1 Answer
Wataru
Sep 7, 2014

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.

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