Math Animated™
Mathematical Introduction for Physics and Engineering
by Samuel Dagan (Copyright © 2007)

Chapter 1: Differentiation; Section 3: Derivatives; page 2

Properties, Exercise 5

Question

$x_1{x}_2\mathrm{.....}{x}_{n-1}{x}_n$ are n given points on the x axis. Find the x that makes

Parts

1. If the expression that should be minimised is denoted by y , then $$\frac{\text{d}y}{\text{d}x}=-2\sum _{k=1}^n\left(x_k-x\right)=2\sum _{k=1}^n\left(x-x_k\right)$$
2. The stationary point should be calculated from $$\sum _{k=1}^n\left(x-x_k\right)=0$$
3. The summation over x only, gives $$\sum _{k=1}^n\left(x\right)=x\sum _{k=1}^{n}1=nx$$
4. and from parts 2 and 3 one obtains $$nx=\sum _{k=1}^n{x}_k$$ which makes the x equal to the arithmetic mean of the x_k 's :
   
   $$x=\frac{{\displaystyle \sum _{k=1}^n}{x}_k}{n}$$
5. From part 1 one obtains $$\frac{{\text{d}}^{2}y}{\text{d}{x}^2}=2\sum _{k=1}^{n}1=2n>0$$ And therefore this is a minimum.

Score

Each part is worth 2 points.