Q. 26 - Calculus

Question

Use optimization techniques to answer the questions in Exercises 25–30.

Find two real numbers x and y whose sum is $36$ and whose product is as small as possible.

Step-by-Step Solution

Answer

The two real numbers are $x=18$ and $y=18$.

Step 1. Given Information.

The sum of two real numbers is $36$ and the product is as small as possible.

Step 2. Form an equation.

From the given information, 

$$\begin{gathered} x+y=36 \\ y=36-x \\ f\left(x\right)=xy \\ =x(36-x) \end{gathered}$$

Step 3. Find the critical points.

$$\begin{gathered} f\left(x\right)=x(36-x) \\ f\text{'}\left(x\right)=\left\{\begin{array}{l}36x+2x,x<0\\ 36x-2x,x\ge 0\end{array}\right. \\ f\text{'}\left(x\right)=0 \\ 36-2x=0 \\ 2x=36 \\ x=18 \end{gathered}$$

Step 4. Find y .

Substitute the value of x in the equation, 

$$\begin{gathered} y=36-x \\ =36-18 \\ =18 \end{gathered}$$