Question

cIf `a_1,a_2,a_3,..,a_n in R` then `(x-a_1)^2+(x-a_2)^2+....+(x-a_n)^2` assumes its least value at x=
A. `a_(1) + a_(2) +....+a_(n)`
B. `2(a_(1) + a_(2), a_(3) +....+a_(n))`
C. `n(a_(1)+a_(2)+....+a_(n))`
D. none of these

Answer 1

Correct Answer - D
We have, `(x-a_(1))^(2)+(x-a_(2))^(2)+....+(x-a_(n))^(2)`
`=nx^(2)-2x(a_(1)+a_(2)+....+a_(n))+(a_(1)^(2)+a_(2)^(2)+....+a_(n)^(2))`
Clearly, `y = nx^(2) - 2x (a_(1)+a_(2) +...+a_(n))+(a_(1)^(2)+a_(2)^(2)+...+a_(n)^(2))` represents a parabola which opens upward. So, it attains its minimum value at the vertex i.e. at
`x = (2(a_(1)+a_(2)+....+a_(n)))/(2n)=(a_(1)+a_(2)+....+a_(n))/(n)["Using x" = (-b)/(2a)]`