Find two positive numbers that satisfy the given requirements. (Enter your answers as a comma-separated list.) The product is 238 and the sum is a minimum.

Question

Find two positive numbers that satisfy the given requirements. (Enter your answers as a comma-separated list.)

The product is 238 and the sum is a minimum.

Answer 1

xy = 238; s = x + y = x + 238/x; s' = 1 - 238/x² = 0; x² = 238; x = √238 and y = 238 - √238

s'' = 476/x³ > 0 at x = √238. So, s has minimum

Answer 2

If xy=238, then y = 238/x.

So the sum of the two numbers can be represented by:

S=x+238x^-1

To find the min, set 1st derivative equal to zero:

dS/dx = 1-238x^-2

1-(238/x²) = 0

238/x²=1

x²=238

x=√(238) (reject the minus square root--problem is to find two positive numbers)

y=238/√(238)=√(238)

Apply the 1st or 2nd derivative test to show that this value of x generates a minimum sum.

S'' = 238x^-3

S''(√238)=√(238)/238 > 0 -- concave up, so a minimum

Check it out:

desmos.com/calculator/yufbdgwfk5

Answer 3