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KVPY Sample Paper KVPY Stream-SX Model Paper-28

  • question_answer
     Suppose that \[f\] is a differentiable function with the property that \[f\left( x+y \right)=f\left( x \right)+f\left( y \right)+xy\]and \[\underset{h\to 0}{\mathop{\lim }}\,\frac{1}{h}f\left( h \right)=3,\]then

    A) f is a linear function

    B) \[f\left( x \right)=3x+{{x}^{2}}\]

    C) \[f\left( x \right)=3x+\frac{{{x}^{2}}}{2}\]

    D) None of these

    Correct Answer: C

    Solution :

    \[f'\left( x \right)=\underset{h\to 0}{\mathop{\operatorname{Lim}}}\,\frac{f\left( x+h \right)-f\left( x \right)}{h}\]
    \[=\underset{h\to 0}{\mathop{Lim}}\,\frac{f\left( x \right)+f\left( h \right)+xh-f\left( x \right)}{h}\]
    \[=\underset{h\to }{\mathop{Lim}}\,\frac{1}{h}f\left( h \right)+x=3+x\]
    Integrating we get \[f\left( x \right)=3x+\frac{{{x}^{2}}}{2}+\text{ }c\]
    Putting \[x=y=0\]in the given equation, we get
    \[f(0)=0\Rightarrow c=0\therefore f(x)=3x+\frac{{{x}^{2}}}{2}\]


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