( a ) f ‗( a ) = 0
( b ) f ‗( a ) < 0 , if x ∈ ( a – h , a ) and f ‗( x ) > 0 , if x ∈ ( a , a + h ), where h is a small but positive quantity .
3 . ( iii ) If f ‗( a ) = 0 but f ‗( x ) does not changes sign in ( a – h , a + h ), for any positive quantity h , then x = a is neither a point of minimum nor a point of maximum .
2 . Second Derivative Test
Let f ( x ) be a differentiable function on an interval I . Let a ∈ I is such that f ―( x ) is continuous at x = a . Then ,
1 . x = a is a point of local maximum , if f ‗( a ) = 0 and f ―( a ) < 0 . 2 . x = a is a point of local minimum , if f ‗( a ) = 0 and f ‖( a ) > 0 .
3 . If f ‗( a ) = f ―( a ) = 0 , but f ‖ ( a ) ≠ 0 , if exists , then x = a is neither a point of local maximum nor a point of local minimum and is called point of inflection .
4 . If f ‗( a ) = f ―( a ) = f ‗‖( a ) = 0 and f iv ( a ) < 0 , then it is a local maximum . And if f iv > 0 , then it is a local minimum .
nth Derivative Test
Let f be a differentiable function on an interval / and let a be an interior point of / such that
( i ) f ‗( a ) = f ―( a ) = f ‗‖( a ) = … f n – 1 ( a ) = 0 and
( ii ) f n ( a ) exists and is non-zero , then If n is even and f n ( a ) < 0 ⇒ x = a is a point of local maximum .
If n is even and f n ( a ) > 0 ⇒ x = a is a point of local minimum .
If n is odd ⇒ x = a is a point of local maximum nor a point of local minimum .
Important Points to be Remembered