® Second Derivative Test Let f be a function defined on an interval I and c ∈ I. Let f be twice differentiable at c . Then (i) x = c is a point of local maxima if f ′ ( c ) = and f ″ ( c ) < The values f ( c ) is local maximum value of f . (ii) x = c is a point of local minima if f ′ ( c ) = and f ″ ( c ) > In this case, f ( c ) is local minimum value of f .
(iii) The test fails if f ′ ( c ) = and f ″ ( c ) = . In this case, we go back to the first derivative test and find whether c is a point of maxima, minima or a point of inflexion.