Let \(A, G, H\) and \(S\) respectively denote the arithmetic mean, geometric mean, harmonic mean and the sum…
Let \(A, G, H\) and \(S\) respectively denote the arithmetic mean, geometric mean, harmonic mean and the sum of the numbers \(a_1, a_2, a_3, \ldots, a_n\). Then the value of \(x\) at which the function \(f(x)=\sum_{k=1}^n\left(x-a_k\right)^2\) has minimum is
\(S\)
\(\mathrm{H}\)
\(G\)
\(A\)
Solution
Given function \(f(x)=\sum_{k=1}^n\left(x-a_k\right)^2\)
\(=\sum_{k=1}^n\left(x^2-2 x a_k+a_k^2\right)\)
\(\begin{aligned}
& =n x^2-2 x\left(a_1+a_2+a_3+\ldots a_n\right) \\
& +\left(a_1^2+a_2^2+\ldots+a_n^2\right)
\end{aligned}\)
\(\because\) The quadratic expression \(a x^2+b x+c\) has its minimum value at \(x=-\frac{b}{2 a}\).
\(\therefore f(x)\) has it's minimum value at
\(\begin{aligned}
& x=-\frac{-2\left(a_1+a_2+a_3+\ldots+a_n\right)}{2 n} \\
& =\frac{a_1+a_2+a_3+\ldots+a_n}{n} \Rightarrow x=A
\end{aligned}\)
Hence, option (4) is correct.