Question

Use optimization techniques to answer the questions in Exercises 25–30.
Find real numbers a and b whose sum is $100$and for which the sum of the squares of a and b is as small as possible.

Short answer

The two real numbers are $a=50$and $b=50$.

Step-by-step solution

Step 1. Given Information.

The sum of two real numbers is $100$ and the sum of the squares is as small as possible.

Step 2. Form an equation.

From the given information,

$$\begin{gathered} a+b=100 \\ b=100-a \\ f\left(x\right)=a^2+b^2 \\ =a^2+{(100-a)}^2 \\ =a^2+10000-200a+a^2 \\ =2a^2+10,000-200a \end{gathered}$$

Step 3. Find the critical points.

$$\begin{gathered} f\left(x\right)=2a^2+10,000-200a \\ f\text{'}\left(x\right)=4a-200 \\ f\text{'}\left(x\right)=0 \\ 4a-200=0 \\ 4a=200 \\ a=50 \end{gathered}$$

Step 4. Find b.

Substitute the value of x in the equation,

$$\begin{gathered} b=100-a \\ =100-50 \\ =50 \end{gathered}$$