CALCULUS WORKSHEET ON OPTIMIZATION Work the following on notebook paper. Write a function for each problem, and justify your answers. Give all decimal answers correct to three decimal places. 1. Find two positive numbers such that their product is 192 and the sum of the first plus three times the second is a minimum.

Step-by-step explanation:

Let the two positive numbers be x and y

If their product is 192, then

xy = 192

If the sum of the first plus three times the second is a minimum this is expressed as;

x + 3y < 0

x < -3y

Substitute into equation 1

xy = 192

-3y(y) = 192

-3y² = 192

y² = 192/-3

y² = -64

Since y is a positive number then;

y² = 64

y = ±√64

y = ±8

Since xy = 192

8x = 192

x = 192/8

x = 24

Hence the two positive values are 8 and 24

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step-by-step explanation: