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JEE Main & Advanced Mathematics Differentiation Question Bank Derivative at a point Standard Differentiation

  • question_answer
     Let \[f(x)\] be a polynomial function of the second degree. If \[f(1)=f(-1)\] and \[{{a}_{1}},{{a}_{2}},{{a}_{3}}\] are in A.P. then \[{f}'({{a}_{1}})\], \[{f}'({{a}_{2}})\], \[{f}'({{a}_{3}})\] are in [AMU 2005]

    A)          A.P

    B)            G.P.

    C)            H.P.

    D)            None of these

    Correct Answer: A

    Solution :

               Let\[f(x)=a{{x}^{2}}+bx+c\]                    Then \[f'(x)=2ax+b\] also, \[f(1)=f(-1)\]                                  \[a+b+c=a-b+c\] Þ b = 0                    \[\therefore \] \[{f}'(x)=2ax\]; \[\therefore \] \[{f}'({{a}_{1}})=2a{{a}_{1}}\]                    \[{f}'(a{{ & }_{2}})=2a{{a}_{2}}\], \[{f}'({{a}_{3}})=2a{{a}_{3}}\]                    As \[{{a}_{1}},{{a}_{2}},{{a}_{3}}\] are in A.P. \[{f}'({{a}_{1}}),\,{f}'({{a}_{2}}),\,{f}'({{a}_{3}})\] are in A.P.


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