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Chapter 4: Problem 11

What two positive real numbers whose product is 50 have the smallest possible sum?

Short Answer

Expert verified
Answer: The two positive real numbers are sqrt(50) and sqrt(50). Their smallest sum is 2*sqrt(50).

Step by step solution

01

Setup the problem

Let x and y be two positive real numbers such that their product is 50. We want to minimize their sum. Write the equation for the product: xy = 50 Now, we want to minimize the sum, which is given by the function: S(x) = x + y
02

Find the second equation

We don't know y explicitly, but we know xy = 50. We will solve for y as a function of x and substitute that into the sum function. y = 50/x Now, the sum function becomes dependent only on x: S(x) = x + 50/x
03

Optimize the sum function

To find the minimum sum, we find the critical points of the function S(x) by taking its first derivative and finding the points where it is equal to zero or does not exist. Calculate the first derivative of S(x) using the power rule and sum rule for derivatives: S'(x) = 1 - 50/x^2
04

Find the critical points

Set the first derivative equal to zero and solve for x: 1 - 50/x^2 = 0 Add 50/x^2 to both sides: 1 = 50/x^2 Multiply both sides by x^2 to get rid of the fraction: x^2 = 50 Take the positive square root of both sides (since x>0): x = sqrt(50)
05

Find the smallest sum

Now substitute the value of x we found, back into our equation for y, then find the value of the sum function: y = 50/x y = 50/sqrt(50) y = sqrt(50) The smallest sum is then: S(x) = x + y S(sqrt(50)) = sqrt(50) + sqrt(50) S(sqrt(50)) = 2*sqrt(50) So, the two positive real numbers whose product is 50 and have the smallest possible sum are sqrt(50) and sqrt(50). Their smallest sum is 2*sqrt(50).

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