JEE Main & Advanced Mathematics Applications of Derivatives Question Bank Rolle's theorem Lagrange's mean value theorem

  • question_answer
    Consider the function \[f(x)={{e}^{-2x}}\]sin 2x over the interval \[\left( 0,\frac{\pi }{2} \right)\]. A real number \[c\in \left( 0,\frac{\pi }{2} \right)\,,\] as guaranteed by Rolle?s theorem, such that \[{f}'\,(c)=0\] is [AMU 1999]

    A)            \[\pi /8\]

    B)            \[\pi /6\]

    C)            \[\pi /4\]

    D)            \[\pi /3\]

    Correct Answer: A

    Solution :

               \[f(x)={{e}^{-2x}}\sin 2x\] Þ \[{f}'(x)=2{{e}^{-2x}}(\cos 2x-\sin 2x)\]            Now,\[{f}'(c)=0\]            Þ\[\cos 2c-\sin 2c=0\]Þ\[\tan 2c=1\]Þ\[c=\frac{\pi }{8}\].

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