optimization

Optimization is the problem of minimizing or maximizing a certain function.

Example Problem 1

Find two positive numbers whose sum is 20 and whose product is as large as posible

$$\begin{array}{rl}x+y& =20⟹y=20-x\\ & \\ & \\ p& =xy\\ & =x(20-x)\\ & =20x-x^2\\ & \\ p^{\prime }& =20-2x=0\\ x& =10\end{array}$$

A first derivative test is necessary to prove that the number is an absolute maximum/minimum.

Example Problem 2

Minimize $S=2x+y$, $xy=288$, $x>=0,y>=0$

$$\begin{array}{rl}y& =\frac{288}{x}\\ S& =2x+\frac{288}{x}\\ S^{\prime }& =2-\frac{288}{x^2}=0\\ x& =12\end{array}$$

Do a first derivative test to confirm the result.