Let’s calculate the derivative first by the quotient rule,
![{\displaystyle f'(x)={\frac {(x-1)'x^{2}-(x^{2})'(x-1)}{x^{4}}}={\frac {x^{2}-2x^{2}+2x}{x^{4}}}={\frac {(2-x)x}{x^{4}}}={\frac {2-x}{x^{3}}}.}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/2bf8325b5aaee8cd43f7de14209d818cd79d8018)
Recall that the domain of
in part (a) is
. On the domain, the only solution of
is
.
Therefore, at the points
, the derivative
might change its sign. So, we make a partition of intervals in the real line based on these points and examine the sign of
;
(1) when
,
, so
;
(2) when
,
, so
;
(3) when
,
, so
.
By the Hint, this tells us that
is increasing in the interval
, while it is decreasing in