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Question
The function f(x) = \[\sum^5_{r = 1}\] (x \[-\] r)2 assumes minimum value at x = ______________ .
Options
5
`5/2`
3
2
Solution
3
\[\text { Given:} f\left( x \right) = \sum^5_{r = 1} \left( x - r \right)^2 \]
\[ \Rightarrow f\left( x \right) = \left( x - 1 \right)^2 + \left( x - 2 \right)^2 + \left( x - 3 \right)^2 + \left( x - 4 \right)^2 + \left( x - 5 \right)^2 \]
\[ \Rightarrow f'\left( x \right) = 2\left( x - 1 + x - 2 + x - 3 + x - 4 + x - 5 \right)\]
\[ \Rightarrow f'\left( x \right) = 2\left( 5x - 15 \right)\]
\[\text { For a local maxima and a local minima, we must have }\]
\[f'\left( x \right) = 0\]
\[ \Rightarrow 2\left( 5x - 15 \right) = 0\]
\[ \Rightarrow 5x - 15 = 0\]
\[ \Rightarrow 5x = 15\]
\[ \Rightarrow x = 3\]
\[\text { Now,} \]
\[f''\left( x \right) = 10\]
\[f''\left( x \right) = 10 > 0\]
\[\text { So, x = 3 is a local minima }. \]
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