And you'll be on the walls of the hall of fame.

Algebra Level 3

If a belongs to real numbers and $a_1 , a_2 , a_3, .\ldots, a_N$ belong to real numbers then $(x - a_1) ^2 + (x-a_2) ^2 + (x - a_3)^2 +\cdots + (x - a_N)^2$ assumes the least value at $x =$

n(a1 + a2+.........aN) 2(a1 + a2+.........aN) 1/n(a1+ a2 + ..........aN) (a1 + a2+.........aN)

1 solution

Tom Engelsman
Oct 30, 2020

It's TGIF, and I'm feelin' lazy with calculus! The above expression sums up to :

$f(x) = Nx^2 - 2(a_{1} + … + a_{N})x + (a^{2}_{1} + … + a^{2}_{N})$ (i),

which has a first derivative:

$f'(x) = 2Nx - 2(a_{1} + … + a_{N}) = 0 \Rightarrow x = \frac{a_{1} + … + a_{N}}{N}$ (ii),

and second derivative:

$f''(x) = 2N > 0$ for all $x \in \mathbb{R}$ (iii).

Thus by (ii) and (iii), the global minimum of $f(x)$ occurs at $\boxed{x =\frac{a_{1} + … + a_{N}}{N}}.$

And you'll be on the walls of the hall of fame.

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