#### Question

The function f(x) = \[\sum^5_{r = 1}\] (x \[-\] r)^{2} assumes minimum value at x =

(a) 5

(b) \[\frac{5}{2}\]

(c) 3

(d) 2

#### Solution

\[(c) 3\]

\[\text { Given:} f\left( x \right) = \sum^5_{r = 1} \left( x - r \right)^2 \]

\[ \Rightarrow f\left( x \right) = \left( x - 1 \right)^2 + \left( x - 2 \right)^2 + \left( x - 3 \right)^2 + \left( x - 4 \right)^2 + \left( x - 5 \right)^2 \]

\[ \Rightarrow f'\left( x \right) = 2\left( x - 1 + x - 2 + x - 3 + x - 4 + x - 5 \right)\]

\[ \Rightarrow f'\left( x \right) = 2\left( 5x - 15 \right)\]

\[\text { For a local maxima and a local minima, we must have }\]

\[f'\left( x \right) = 0\]

\[ \Rightarrow 2\left( 5x - 15 \right) = 0\]

\[ \Rightarrow 5x - 15 = 0\]

\[ \Rightarrow 5x = 15\]

\[ \Rightarrow x = 3\]

\[\text { Now,} \]

\[f''\left( x \right) = 10\]

\[f''\left( x \right) = 10 > 0\]

\[\text { So, x = 3 is a local minima }. \]