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

1 Answer
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.