Question

Let $A, G, H$ and $S$ respectively denote the arithmetic mean, geometric mean, harmonic mean and the sum of the numbers $a_1 , a_2 , a_3 ....., a_n$ . Then the value of at which the function $f(x) =\displaystyle \sum^n_{k =1} (x -a_k)^2$ has minimum is

• A. S
• B. H
• C. G
• D. A

Answer 1

The Correct Option is D

Given function $f(x)=\displaystyle \sum_{k=1}^{n}\left(x-a_{k}\right)^{2}$ $=\displaystyle \sum_{k=1}^{n}\left(x^{2}-2 x a_{k}+a_{k}^{2}\right)$ $=n x^{2}-2 x\left(a_{1}+a_{2}+a_{3}+\ldots a_{n}\right)$ $+\left(a_{1}^{2}+a_{2}^{2}+\ldots+a_{n}^{2}\right)$ $\because$ 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 $x=-\frac{-2\left(a_{1}+a_{2}+a_{3}+\ldots+a_{n}\right)}{2 n}$ $=\frac{a_{1}+a_{2}+a_{3}+\ldots+a_{n}}{n} \Rightarrow x=A$