Q109 / 160 · AP EAMCET 2019 (20 Apr Shift 1) · Mathematics

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

  • A
    \(S\)
  • B
    \(\mathrm{H}\)
  • C
    \(G\)
  • Correct answer
    D
    \(A\)

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.

Hence, the correct answer is option: 'D'.

More Sequences and Series Questions