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JEE Main & Advanced Mathematics Applications of Derivatives Question Bank Increasing and Decreasing Function

  • question_answer
    Function \[f(x)=\frac{\lambda \sin x+6\cos x}{2\sin x+3\cos x}\] is monotonic increasing, if                             [MP PET 2001]

    A)            \[\lambda >1\]

    B)            \[\lambda <1\]

    C)            \[\lambda <4\]

    D)            \[\lambda >4\]

    Correct Answer: D

    Solution :

               The function is monotonic increasing, if \[{f}'(x)>0\]            Þ \[\frac{(2\sin x+3\cos x)\,(\lambda \cos x-6\sin x)}{{{(2\sin x+3\cos x)}^{2}}}\] \[-\frac{(\lambda \sin x+6\cos x)(2\cos x-3\sin x)}{{{(2\sin x+3\cos x)}^{2}}}>0\]            Þ \[3\lambda ({{\sin }^{2}}x+{{\cos }^{2}}x)-12({{\sin }^{2}}x+{{\cos }^{2}}x)>0\]            Þ \[3\lambda -12>0\] Þ \[\lambda >4.\]


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