KVPY Sample Paper KVPY Stream-SX Model Paper-25

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, \[f\left( x \right)=\sum\limits_{i=1}^{n}{{{\left( x-{{a}_{i}} \right)}^{2}}}\] \[\Rightarrow \]\[f'(x)=\sum\limits_{i=1}^{n}{2(x-{{a}_{i}})}\]

\[=2nx-({{a}_{1}}+{{a}_{2}}.....+{{a}_{n}})\]

\[=2nx-2nA\]where \[A=\frac{{{a}_{1}}+{{a}_{2}}+....+{{a}_{n}}}{n}\]

\[f'\left( x \right)=0\Rightarrow x=\operatorname{A}.\,Again\,f''\left( x \right)=2n>0\]