Final answer:
The two positive real numbers whose product is 67 and that have the smallest possible sum are approximately 8.185 and 8.185, as found by taking the square root of 67, pursuant to the Arithmetic Mean-Geometric Mean Inequality.
Step-by-step explanation:
The question is asking for two positive real numbers whose product is 67 and that have the smallest possible sum. According to the Arithmetic Mean-Geometric Mean Inequality (AM-GM Inequality), for any set of non-negative real numbers, the arithmetic mean is greater than or equal to the geometric mean.
This implies that the smallest sum is achieved when the two numbers are equal since the arithmetic mean and the geometric mean will be the same at this point.
We can find the two numbers by taking the square root of 67 since we want both numbers to be the same (x * x = 67). The square root of 67 is the number that, when multiplied by itself, gives 67.
Step by Step Solution:
- Write down the product equation: x * x = 67.
- Take the square root on both sides: x = sqrt(67).
- Find the square root of 67 using a calculator to get the two numbers, which will be approximately 8.185 and 8.185.
- Confirm that their sum is minimal.
Therefore, the two positive real numbers whose product is 67 and have the smallest possible sum are approximately 8.185 and 8.185.