How can I find two positive numbers whose product is 100 and whose sum is the smallest possible?

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To solve the problem of finding two positive numbers whose product is 100 and whose sum is minimized, we can approach it using algebraic methods. Let’s denote the two numbers as x and y.

From the problem, we know:

  • Product: x * y = 100
  • We want to minimize the sum: S = x + y

We can express y in terms of x using the product equation:

y = 100 / x

Next, we can substitute this expression for y into the sum equation:

S = x + (100 / x)

Now, we need to minimize the function S(x) = x + 100/x. To do this, we can use calculus:

Step 1: Take the derivative of S

Find the first derivative of S:

S'(x) = 1 – 100/x^2

Step 2: Set the derivative to zero

To find critical points, set the derivative equal to zero:

1 – 100/x^2 = 0

This simplifies to:

100/x^2 = 1 → x^2 = 100

Taking the square root gives:

x = 10 (since we only consider positive values)

Step 3: Find y

Substituting x = 10 back into the expression for y:

y = 100 / 10 = 10

Step 4: Confirm the sum

Now we find the sum:

S = 10 + 10 = 20

So, the two positive numbers that satisfy the conditions of the problem are 10 and 10, and their sum, which is minimized, equals 20.

Conclusion

Thus, the solution to the problem is:

  • Numbers: 10 and 10
  • Product: 100
  • Minimum Sum: 20

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