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

  • question_answer
    The maximum value of exp \[(2+\sqrt{3}\cos x+\sin x)\] is [AMU 1999]

    A) \[\exp (2)\]

    B) \[\exp (2-\sqrt{3})\]

    C) \[\exp (4)\]

    D) 1

    Correct Answer: C

    Solution :

    • Let \[y=\exp (2+\sqrt{3}\cos x+\sin x)\]           
    • Þ \[{y}'=\exp (2+\sqrt{3}\cos x+\sin x)\,(-\sqrt{3}\sin x+\cos x)\]           
    • Now \[{y}'=0\] Þ \[-\sqrt{3}\sin x+\cos x=0\]           
    • Þ \[\sin \left( x-\frac{\pi }{6} \right)=0\] Þ \[x=\frac{\pi }{6}\]           
    • Now \[{y}''\] is ?ve at \[x=\frac{\pi }{6}\]           
    • \Maximum value of                   
    • \[y=\exp \,\left( 2+\sqrt{3}\left( \frac{\sqrt{3}}{2} \right)+\frac{1}{2} \right)\]= \[\exp (4)\].


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