Optimization Problem

**abclarinetuvwxyz** · January 6th 2009, 03:41 PM

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.

**DeMath** · January 6th 2009, 04:08 PM

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?

**Last_Singularity** · January 6th 2009, 04:17 PM

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 $x$

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