# Finding the unique point of relative minimum

Let $a_1, a_2, ... a_n$ be real numbers and let $f$ be defined on $\Bbb R$ by $\sum_{i=1}^{n} (a_i -x)^2 , x \in \Bbb R$. A friend and I have been struggling with this for half an hour now. If someone could help us take the derivative of this (or help us determine the final sum), that would be great.

• Remember that differentiation is linear. So you can differentiate each term in the sum individually. – carmichael561 Feb 18 '16 at 2:31

## 1 Answer

There are a number of ways to do this.

First:

$\begin{array}\\ \frac{d}{dx}\sum_{i=1}^{n} (a_i -x)^2 &=\sum_{i=1}^{n} \frac{d}{dx}(a_i -x)^2\\ &=\sum_{i=1}^{n} (-2)(a_i -x)\\ &=-2\sum_{i=1}^{n} (a_i -x)\\ &=-2\sum_{i=1}^{n} a_i +2\sum_{i=1}^{n}x\\ &=-2\sum_{i=1}^{n} a_i +2nx\\ \end{array}$

Second:

$\begin{array}\\ \frac{d}{dx}\sum_{i=1}^{n} (a_i -x)^2 &=\frac{d}{dx}\sum_{i=1}^{n} (a_i^2-2a_ix +x^2)\\ &=\sum_{i=1}^{n} \frac{d}{dx}(a_i^2-2a_ix +x^2)\\ &=\sum_{i=1}^{n} (-2a_i +2x)\\ &=-2\sum_{i=1}^{n} a_i +2nx\\ \end{array}$

Third:

$\begin{array}\\ \frac{d}{dx}\sum_{i=1}^{n} (a_i -x)^2 &=\frac{d}{dx}\sum_{i=1}^{n} (a_i^2-2a_ix +x^2)\\ &=\frac{d}{dx}\left(\sum_{i=1}^{n} a_i^2-\sum_{i=1}^{n}2a_ix +\sum_{i=1}^{n}x^2\right)\\ &=\frac{d}{dx}\left(\sum_{i=1}^{n} a_i^2-2x\sum_{i=1}^{n}a_i +nx^2\right)\\ &=\frac{d}{dx}\sum_{i=1}^{n} a_i^2-2\frac{d}{dx}x\sum_{i=1}^{n}a_i +\frac{d}{dx}nx^2\\ &=-2\sum_{i=1}^{n}a_i +2nx\\ \end{array}$

In all three cases (fortunately), the derivative is zero when $x = \frac1{n}\sum_{i=1}^{n}a_i$, the average of the $a_i$.