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KVPY Sample Paper KVPY Stream-SX Model Paper-2

  • question_answer
    If \[f\,(x)\] is a monotonically decreasing function and have concavity up then its inverse \[{{f}^{-1}}(x)\] will be

    A) decreasing and have concavity upwards

    B) decreasing and have concavity downwards

    C) increasing and have concavity downwards

    D) increasing and have concavity upwards

    Correct Answer: A

    Solution :

    Let \[y=f\,(x)\]
    \[\therefore \]      \[{{f}^{-1}}(y)=x\]
                \[{{f}^{-1}}'\,(y)\,\,.\,\,y'=1\]
                \[{{f}^{-1}}''\,(y)=-\frac{y''}{{{(y')}^{2}}}\]   
    \[\because \]       \[y'<0\] & \[y''>0\]         
    \[\therefore \]      \[{{f}^{-1}}'\,(y)<0\] & \[{{f}^{-1}}''\,(y)>0\]


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