]> Exercise 5

# Properties, Exercise 5

## Question

${x}_{1\text{\hspace{0.28em}}}{x}_{2\text{\hspace{0.28em}}}.....\text{\hspace{0.28em}}{x}_{n-1}\text{\hspace{0.28em}}{x}_{n}$ are n given points on the x axis. Find the x that makes

minimum!

## 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 xk 's :

$x=\frac{\sum _{k=1}^{n}{x}_{k}}{n}$
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.