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KVPY Sample Paper KVPY Stream-SX Model Paper-9

  • question_answer
      Let \[f:\left[ 0,\infty )\to \left[ 0,\infty ) \right. \right.\]and \[g:\left[ 0,\infty )\to \left[ 0,\infty ) \right. \right.\]be non-increasing and non-decreasing functions respectively and \[h(x)\,=g(f(x)).\]if \[f\]and \[g\]are differentiable for all points in their respective domains and \[h(0)=0,\]than

    A) \[h(x)\,=0\forall x\ge 0\]

    B) \[h(x)\,>0\forall x\ne 0\]

    C) \[h(x)\,<0\forall x\ne 0\]

    D) none of these  

    Correct Answer: A

    Solution :

    Given \[h(x)\]=\[g(f(x))\].
    Since\[0\le g(x)<\infty \forall \,x\]
    \[\therefore \,h(x)\ge 0\forall \,x\in domain..\left( i \right)\]    
    Again \[h'\left( x \right)=g'(f(x))\,\,f'\left( x) \right)\le 0\]
    \[\Rightarrow h(x)\,\]is decreasing function
    \[\therefore h\left( x \right)\le h(0)if\,x\ge 0\,\]
    \[\Rightarrow h\left( x \right)\le 0\forall x\ge 0\]
    \[\left( \because h(0)=0 \right)..\left( ii \right)\]
    \[\therefore \]From (i) and (ii) \[h(x)\,=0\forall x\in [0,\infty )\]          


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