# Optimization Problem

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• Jan 6th 2009, 04:41 PM
abclarinetuvwxyz
Optimization Problem
Can someone please help me set up this problem?

Find the larger of two numbers whose sum is 30 for which the sum of their squares is a minimum.
• Jan 6th 2009, 05:08 PM
DeMath
Let the first number $x_1$ and the second number $x_2$.

Then $f\left( {x_1 ,x_2 } \right) = x_1^2 + x_2^2 \to \min$ and $x_1+x_2=30\Rightarrow$.

$\Rightarrow x_2=30-x_1\Rightarrow f\left( {x_1 } \right) = x_1^2 + \left( {30-x_1} \right)^2 \to \min$

Do you undestand?
• Jan 6th 2009, 05:17 PM
Last_Singularity
Quote:

Originally Posted by abclarinetuvwxyz
Can someone please help me set up this problem?

Find the larger of two numbers whose sum is 30 for which the sum of their squares is a minimum.

Denote these two numbers to be $x$ and $y$.

#1: Their sum must be 30, so: $x+y=30$, or $y=30-x$

#2: And you want to minimize: $x^2+y^2$

Given #1 as a constraint, #2 becomes: $x^2+(30-x)^2$. Differentiate this expression, set it equal to zero (to minimize), and find $x$