KVPY Sample Paper KVPY Stream-SX Model Paper-25

  • question_answer
    If \[A\]denotes the arithmetic mean of the real numbers \[{{a}_{1}},{{a}_{2}},...{{a}_{n}},\] then \[\sum\limits_{i=1}^{n}{{{\left( x-{{a}_{i}} \right)}^{2}}}\]has a minimum at.

    A) \[x=0\]

    B) \[x=1\]

    C) \[x=A\]

    D) \[x=A+1\]

    Correct Answer: C

    Solution :

    Let, \[f\left( x \right)=\sum\limits_{i=1}^{n}{{{\left( x-{{a}_{i}} \right)}^{2}}}\] \[\Rightarrow \]\[f'(x)=\sum\limits_{i=1}^{n}{2(x-{{a}_{i}})}\]
    \[=2nx-2nA\]where \[A=\frac{{{a}_{1}}+{{a}_{2}}+....+{{a}_{n}}}{n}\]
    \[f'\left( x \right)=0\Rightarrow x=\operatorname{A}.\,Again\,f''\left( x \right)=2n>0\]

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