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JEE Main & Advanced Sample Paper JEE Main - Mock Test - 9

  • question_answer
    Let \[f:[1,3]\ to [0,\infty )\] be continuous and differentiable function. If \[\left( f(3)-f(1) \right).\]\[\left( {{f}^{2}}(3)+{{f}^{2}}(1)+f(3)f(1) \right)=k{{f}^{2}}(c)f'(c),\] where \[c\in (1,3),\] then the value of k is

    A) \[3\]            

    B) \[6\]         

    C) \[9\]                     

    D) \[12\]

    Correct Answer: B

    Solution :

    [b] Let \[F(x)={{f}^{3}}(x)\] And \[F(x)\] is continuous and differentiable function in \[[1,\,\,3]\]. Using LMVT, we get \[\frac{F(3)-F(1)}{3-1}=F'(c)\] \[\therefore \,\,\,\frac{{{f}^{3}}(3)-{{f}^{3}}(1)}{2}=3{{f}^{2}}(c).f'(c)\] \[\Rightarrow \,\,\,\,\,k=6\]    


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