Optimization !!!!

Question

Find the two positive numbers such that the product is 144 and the sum of twice first and five times the square of the second is minimum??

Answer 1

Let x be the first number.

Then 144-x = second number

Minimize S = 2x + 5(144 - x)²

= 2x + 5(20736 - 288x + x²)

= 5x² - 1438x + 103680

The graph of S is a parabola opening upward.

Minimum occurs when x = -(-1438) / [2(5)] = 143.8

The numbers are 143.8 and 0.2.