Math Help Forum - View Single Post

Krahl · #18 July 28th, 2009, 12:11 PM

This is how i did it...

let S= $|x-2|+|x-2^2|+...+|x-2^8|+|x-2^9|+....+|x-2^{15}|+|x-2^{16}|$

and S= $|x-2^{16}|+....+|x-2^9|+|x-2^8|+.......+|x-2|$

so whatever value minimises S would minimise

2S= $(|x-2|+|x-2^{16}|)+(|x-2^2|+|x-2^{15}|)$
$+......+(|x-2^8|+|x-2^9|)+(|x-2^9|+|x-2^8|)+......+(|x-2^{16}|+|x-2|)$

So if we look at all the values of x which minimises each term then the intersection of the values of x will minimise 2S and S.

looking at the graph of each term as chisigma did we see that all values between the minimum of each of the two terms are minimising values for both terms. since they are linear functions they intersect in the middle.

so the valuse minimising $(|x-2|+|x-2^{16}|)$ are $2 \leq x \leq 2^{16}$ and so on.

the intersection of $2 \leq x \leq 2^{16},2^4 \leq x \leq 2^{15},...,2^8 \leq x \leq 2^9,...,2 \leq x \leq 2^{16}$

is $2^8 \leq x \leq 2^9$ so these values would all minimise S.