Thread: 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.

Let the first number $\displaystyle x_1$ and the second number $\displaystyle x_2$.

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

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

Do you undestand?

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 $\displaystyle x$ and $\displaystyle y$.

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

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

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