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KVPY Sample Paper KVPY Stream-SX Model Paper-16

  • question_answer
    If \[g(x)\] is a differential real valued function satisfying \[g''(x)-3g'(x)>3\,\,\forall \,\,x\,\,\ge 0\,\] and \[g'(0)=-\,1,\] then \[\frac{d}{dx}(g'(x){{e}^{-\,3x}})>3.\,{{e}^{-\,3x}}\] is-

    A) An increasing function

    B) A decreasing function

    C) A constant function

    D) Data insufficient

    Correct Answer: A

    Solution :

    \[\frac{d}{dx}(g'(x){{e}^{-\,3x}})>3.\,{{e}^{-\,3x}}\]
    \[\frac{d}{dx}(g'(x){{e}^{-\,3x}}+{{e}^{-\,3x}})>0\]
    \[\Rightarrow {{e}^{-3x}}(1+g'(x))\] is an increasing function.
    Now, \[{{e}^{-3x}}(1+g'(x))>(g'(0)+1)\]
    \[\Rightarrow x>0\]
    \[\Rightarrow g'(x)+1>0\]
    \[\Rightarrow g(x)+x\]is an increasing function.


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