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KVPY Sample Paper KVPY Stream-SX Model Paper-17

  • question_answer
    Let\[f\] be a real valued function satisfying \[f(x)+f(x+6)=f(x+3)+f(x+9).\] Then \[\int\limits_{x}^{x+12}{f(t)dt}\] is

    A) A linear function of x

    B) An exponential function of x

    C) A constant function

    D) None of these

    Correct Answer: C

    Solution :

    given, \[\,f(x)+f(x+6)=f(x+3)+f(x+9)\]
    Put \[x=x+3\]
    \[f(x+3)+f(x+9)=f(x+6)+f(x+12)\]
    \[\Rightarrow \]\[f(x)=f(x+12)\]
    Let \[g,(x)=\int\limits_{x}^{x+12}{f(t)dt}\]
    \[\Rightarrow \]\[g'(x)=f(x+12)-f(x)=0\]
    \[\Rightarrow \]\[g(x)\]is a constant function.


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