Theorem 4 (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) = 0 and f ″(c) < 0
The value f (c) is local maximum value of  f (ii)
x = c is a point of local minima if 
( )
0
f
′c =
 and f ″(c) > 0
In this case, f (c) is local minimum value of  f