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JEE Main & Advanced AIEEE Solved Paper-2003

  • question_answer
    Let \[f(x)\] be a polynomial function of second degree. If \[f(1)=f(-1)\] and a, b, c are in AP, then \[f'\,(a),\,\,f'(b)\] and \[f'(c)\] are in     AIEEE  Solved  Paper-2003

    A) AP                         

    B) GP

    C) HP

    D) arithmetico-geometric progression

    Correct Answer: A

    Solution :

    Let \[f(x)=A{{x}^{2}}+B{{x}^{2}}+C\] \[\therefore \]      \[f(1)=A+B+C\] and            \[f(-1)=A-B+C\] \[\because \]         \[f(1)=f(-1)\] \[\Rightarrow \]   \[A+B+C=A-B+C\] \[\Rightarrow \]   \[2B=0\] \[\Rightarrow \]   \[B=0\]     \[\therefore f(x)=A{{x}^{2}}+C\] Now,         \[f'(x)=2Ax\] \[\therefore f'(a)=2\,Aa,\,f'(b)=2Ab,f'(c)=2Ac\] Also. a, b. c are in AP.             \[\therefore 2\,Aa,\,2\,Ab,\,2\,Ac\] are in AP. Hence, \[f'(a),\,f'(b)f'(c)\] are in AP.


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