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JEE Main & Advanced Mathematics Applications of Derivatives Question Bank Critical Thinking

  • question_answer
    Let \[f(x)=\left\{ \begin{align}   & {{x}^{\alpha }}\ln x,x>0 \\  & 0,\,\,\,\,\,\,\,\,\,\,\,\,x=0 \\ \end{align} \right\}\], Rolle?s theorem is applicable to f for \[x\in [0,1]\], if \[\alpha =\] [IIT Screening 2004]

    A) - 2

    B) - 1

    C) 0

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

    Correct Answer: D

    Solution :

    • For Rolle?s theorem to be applicable to f, for \[x\in [0,\,1]\], we should have (i) \[f(1)=f(0)\],              
    • (ii) f is continuous for \[x\in [0,\,1]\] and f is differentiable for \[x\in (0,\,1)\]              
    • From (i), \[f(1)=0\], which is true.               
    • From (ii), \[0=f(0)=f({{0}_{+}})=\underset{x\to {{0}_{+}}}{\mathop{\lim }}\,{{x}^{\alpha }}\ln x\] Which is true only for positive values of \[\alpha \], thus  is correct.


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