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JAMIA MILLIA ISLAMIA Jamia Millia Islamia Solved Paper-2011

  • question_answer
        The value of C in Lagranges theorem for the function\[f(x)=log\text{ }sin\text{ }x\]in the interval\[\left[ \frac{\pi }{6},\frac{5\pi }{6} \right]\]is

    A)  \[\frac{\pi }{4}\]                                             

    B)  \[\frac{\pi }{2}\]

    C)  \[\frac{2\pi }{3}\]                           

    D)  None of these

    Correct Answer: B

    Solution :

                    \[f\left( \frac{5\pi }{6} \right)=\log \sin \left( \frac{5\pi }{6} \right)=\log \sin \frac{\pi }{6}=\log \frac{1}{2}\] \[=-\log 2\]                 \[f\left( \frac{\pi }{6} \right)=\log \sin \left( \frac{\pi }{6} \right)=-\log 2\] Now,     \[f(x)=\frac{1}{\sin x}\cos x=\cot x\] By LMV theorem,                 \[\frac{f\left( \frac{5\pi }{6} \right)-f\left( \frac{\pi }{6} \right)}{\left( \frac{5\pi }{6} \right)-\left( \frac{\pi }{6} \right)}=\cot C\] \[\Rightarrow \]               \[\cot C=0\]        \[\Rightarrow \]               \[C=\frac{\pi }{2}\in \left( \frac{\pi }{6},\frac{5\pi }{6} \right)\]


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