of inflection of f x ( ) . Theorem . (Test for Points of Inflection) (i) If ′′ c ( ) exists and ′′ c ( ) changes sign when passing through x , then the point ( , ( )) c f c is a point of inflection of the graph of f . (ii) If ′′ c ( ) exists at the point of inflection, then ′′ c ( ) .
Remark To determine the position of points of inflexion on the curve y ( ) it is necessary to find the points where ′′ ( ) changes sign. For ‘smooth’ curves (no sharp corners), this may happen when either (i) ′′ or (ii) ′′ ( ) does not exist at the point. Remark ( ) It is also possible that ′′ c ( ) may not exist, but ( , ( )) c f c could be a point of inflection. For instance, f x ( ) = at c = .
( ) It is possible that ′′ c ( ) at a point but ( , ( )) c f c need not be a point of inflection. For instance, f x ( ) = at c = . ( ) A point of inflection need not be a stationary point. For instance, if f x then, ′ and ′′ = − and hence ( , ) p is a point of inflection but not a stationary point for f x ( ) .
Example . Determine the intervals of concavity of the curve f x ), ⋅ and, points of inflection if any. The given function is a polynomial of degree . Now, ′ f ( ) = ( ⋅ = ⋅ ′′ ( ) = (( ) ( )) ⋅ = ) ( ⋅ Now, ′′ ( ) = ⇒ Fig.
. ′′ x ( ) - - The intervals of concavity are tabulated in Table . . Interval (-∞, ) ( , ) ( , ∞)