Let f ( x ) be a polynomial function of second degree. If f ( 1 ) = f ( − 1 ) and a , b , c are in A.P, then f ′ ( a ) , f ′ ( c ) are in

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f ( x ) = a x 2 + b x + c f ( 1 ) = f ( − 1 ) ⇒ a + b + c = a − b + c or b = 0 ∴ f ( x ) = a x 2 + c or f ′ ( x ) = 2 a x Now f ′ ( a ) ; f ′ ( b ) ; and f ′ ( c ) are 2 a ( a ) ; 2 a ( b ) ; 2 a ( c ) . If a , b , c are in A.P. then f ′ ( a ) ; f ′ ( b ) and f ′ ( c ) are also in A.P.

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