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KVPY Sample Paper KVPY Stream-SX Model Paper-1

  • question_answer
    Suppose \['f'\] is continuous function from \[R\,\,\]to \[R\,\,\] and \[f(f(a))=a\] for some \[a\in R\] then the equation \[f\left( x \right)=x\] has

    A) No solution

    B) Exactly one solution

    C) At most one solution

    D) Atleast three solutions

    Correct Answer: D

    Solution :

    (d)
    If \[f\left( a \right)=a,\]then obviously \[x=a\]is the solution
    Let \[f\left( a \right)>a\]and \[g\left( x \right)=f\left( x \right)-x\]then \[g\left( a \right)>0\]and \[g\left( f\left( a \right) \right)=f\left( f\left( a \right) \right)-f\left( a \right)=a-f\left( a \right)<0\]
    Since\[~g\left( x \right)~\]is continuous, so at least for one \[c\in \left( a,f\left( a \right) \right),g\left( c \right)=0.\]
    Similarly we can argue for\[f\left( a \right)<a\]
     


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