Question

The expression $f\left(x\right)=\sum _5^{k=1}(x-k{)}^2$ assumes minimum values for $x$ given by

• A. $3$
• B. $2$
• C. $\frac{5}{2}$
• D. $5$

Answer 1

Answer: (A)

$3$

$f\left(x\right)={\sum }_{k=1}^5(x-k{)}^2$

$\therefore f\left(x\right)=(x-1{)}^2+(x-2{)}^2+(x-3{)}^2+(x-4{)}^2+(x-5{)}^2$

$\therefore f\left(x\right)=5x^2-30x+55$

$\therefore f\left(x\right)=5(x^2-6x+11)$

$\therefore f\left(x\right)=5(x^2-6x+9+2)$

$\therefore f\left(x\right)=5(x^2-6x+9)+10$

$\therefore f\left(x\right)=5(x-3{)}^2+10$

So, $f\left(x\right)$ assumes minimum value at $x=3$