( 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