Example 21 Find all the points of local maxima and local minima of the function f
given by
f(x) = 2x3 – 6x2 + 6x +5 Solution We have
f(x) = 2x3 – 6x2 + 6x +5
or
2
2
( )
6
12
6
6(
1)
( )
12(
1)
f
x
x
x
x
f
x
x
 ′
=
−
+
=
−
 ′′
=
−

Now f ′(x) = 0 gives x =1 Also f ″(1) = 0 Therefore, the second derivative test
fails in this case