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JCECE Engineering JCECE Engineering Solved Paper-2010

  • question_answer
    If \[f(x)\] is differentiable in the interval \[[2,\,\,5]\] where\[f(2)=\frac{1}{5}\]and\[f(5)=\frac{1}{2}\], then there exist a number \[c,\,\,2<c<5\] for which \[f'(c)\] is equal to

    A) \[\frac{1}{2}\]                                   

    B) \[\frac{1}{5}\]

    C) \[\frac{1}{10}\]                                

    D)  None of these

    Correct Answer: C

    Solution :

    As \[f(x)\] is differentiable in-[2, 5], therefore it is also continuous in\[[2,\,\,5]\]. Hence, by mean value theorem, there exist a real number \[c\] in \[(2,\,\,5)\] such that                 \[f'(c)=\frac{f(5)-f(2)}{5-2}\] \[\Rightarrow \]               \[f'(c)=\frac{\frac{1}{2}-\frac{1}{5}}{3}=\frac{1}{10}\]                 \[\left[ \because \,\,f(2)=\frac{1}{5}\,\,and\,\,f(5)=\frac{1}{2}\,\,are\,\,given \right]\]


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