# Thread:

1. ## Help me please

Hello,
can any one help me in solving the folowing problem please???

show that if $x_1,...,x_n$ is a set of real numbers
such that
$<br /> \sum_{i=1,..N} {x_i=a}<br />$
then the quantity
$\sum_{i=1,..N} {x_i^2}$
is minimized when $x_1=x_2=...=x_n=a/n$

the problem give some hint for solution:
assume that $x_i != x_j$
and show that this contradics the assumption that
$x_1,...,x_n$ minimize the sum of squares.to do this show that replacing $x_1 and x_j$ each by
$(x_i+x_j)/2$ reduces the sum of squares while insuring
$<br /> x_1+...+x_i_-_1+(x_i+x_j)/2+x_i_+_1$ $+...+x_j_-_1+(x_i+x_j)/2+x_j_+_1+...+x_n=a$
thanks.

2. Originally Posted by TFT
Hello,
can any one help me in solving the folowing problem please???

show that if $x_1,...,x_n$ is a set of real numbers
such that
$<br /> \sum_{i=1,..N} {x_i=a}<br />$
then the quantity
$\sum_{i=1,..N} {x_i^2}$
is minimized when $x_1=x_2=...=x_n=a/n$
You want to minimize,
$f(x_1,...,x_n)=x_1^2+x_2^2+...+x_n^2$
With constraint curve,
$C(x_1,x_2,...,x_n)=x_1+...+x_n=a$
Use Lagrange multipliers,
$\nabla f=k\nabla g$
Thus,
$2x_1(1,0,...,0)+2x_2(0,1,...,0)+...+2x_n(0,0,...,1 )$ $=k(1,0,...,0)+k(0,1,...,0)+...+k(0,0,...,1)$
From here we see that,
$x_1=x_2=...=x_n=k/2$
Thus,
$x_1=...=x_n=\frac{a}{n}$

Take point $(a,0,...,0)$ and note that $f(a,0,...,0)\geq f(x_1,x_2,...,x_n)$. Therefore, the point we found was the minimum!
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 23, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.8720973134040833, "perplexity_flag": "middle"}