Let A, G, H and S respectively denote the ari | vedclass

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(D) Given the function $f(x) = \sum_{k=1}^n (x - a_k)^2$. Expanding the summation,we get $f(x) = \sum_{k=1}^n (x^2 - 2xa_k + a_k^2)$. This simplifies

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(D) Given the function $f(x) = \sum_{k=1}^n (x - a_k)^2$. Expanding the summation,we get $f(x) = \sum_{k=1}^n (x^2 - 2xa_k + a_k^2)$. This simplifies to $f(x) = nx^2 - 2x \sum_{k=1}^n a_k + \sum_{k=1}^n a_k^2$. Since $f(x)$ is a quadratic expression of the form $ax^2 + bx + c$,it attains its minimum value at $x = -\frac{b}{2a}$. Here,$a = n$ and $b = -2 \sum_{k=1}^n a_k$. Thus,the minimum occurs at $x = -\frac{-2 \sum_{k=1}^n a_k}{2n} = \frac{\sum_{k=1}^n a_k}{n}$. By definition,the arithmetic mean $A = \frac{\sum_{k=1}^n a_k}{n}$. Therefore,$x = A$.

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 $f(x)=\sum_{k=1}^n(x-a_k)^2$ has a minimum is

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