1. ## Optimization Problem

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

2. 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?

3. Originally Posted by abclarinetuvwxyz
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 $x$