Let (A, G, H ) and (S ) respectively denote the arithmetic mean, geometric mean, harmonic mean and the sum of the numbers (a₁, a₂, a₃, , a_n ). Then the value of (x ) at which the function (f(x)= _ k=1 ^n (x-a_k )^2 ) has minimum is

Step-by-step solution

Given function (f(x)= _ k=1 ^n (x-a_k )^2 ) (= _ k=1 ^n (x^2-2 x a_k+a_k^2 ) ) ( aligned & =n x^2-2 x (a₁+a₂+a₃+ a_n ) & + (a₁^2+a₂^2+ +a_n^2 ) aligned ) ( ) The quadratic expression (a x^2+b x+c ) has its minimum value at (x=- b 2 a ). ( f(x) ) has it's minimum value at ( aligned & x=- -2 (a₁+a₂+a₃+ +a_n ) 2 n & = a₁+a₂+a₃+ +a_n n x=A aligned ) Hence, option (4) is correct.