Then
 (i) If f ′(x) changes sign from positive to negative as x increases through c,
i e , if f ′(x) > 0 at every point sufficiently close to and to the left of c,
and f ′(x) < 0 at every point sufficiently close to and to the right of c,
then c is a point of local maxima Rationalised 2023-24
 MATHEMATICS
186
(ii)
If f ′(x) changes sign from negative to positive as x increases through c,
i