the sum of tow postive real number is 20
the minum possible values or the sum of the cube is
The minimum possible value for the sum of the cubes can be found by using AM-GM inequality.
By AM-GM inequality, we have:
(a³ + b³) / 2 ≥ √(a³ * b³)
Since the sum of the two positive real numbers is 20, let's assume the numbers are a and b.
(a + b) = 20
Rearranging the equation to solve for a, we get a = 20 - b.
Substituting this value of a into the inequality, we have:
((20 - b)³ + b³) / 2 ≥ √((20 - b)³ * b³)
Expanding and simplifying:
(8000 - 120b + 6b²) / 2 ≥ √((20 - b)³ * b³)
4000 - 60b + 3b² ≥ √((20 - b)³ * b³)
Squaring both sides to get rid of the square root:
(4000 - 60b + 3b²)² ≥ (20 - b)³ * b³
Expanding and simplifying further:
16000000 - 960000b + 24000b² - 3600b³ + 108b⁴ - 9b⁶ ≥ (20 - b)³ * b³
Simplifying:
108b⁴ - 9b⁶ + 24000b² - 3600b³ - 960000b + 16000000 ≥ 0
Now, we need to find the values of a and b that minimize this equation. However, solving this equation is complex and requires the use of numerical methods like calculus or graphing.
To find the minimum possible value for the sum of the cubes of two positive real numbers whose sum is 20, we can use the concept of AM-GM inequality.
According to the AM-GM inequality, for any two positive real numbers a and b, the sum of their cubes will be minimum when both numbers are equal.
So, let's assume the two positive real numbers as x and y. We know that x + y = 20.
To find the minimum possible value of x^3 + y^3, we need to find the minimum value of (x + y)^3.
Expanding (x + y)^3 using the binomial theorem, we get:
(x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3
Since x + y = 20, we can rewrite (x + y)^3 as (20)^3:
(x + y)^3 = 20^3 = 8000
Now, let's compare the two expressions (x^3 + y^3) and 8000:
x^3 + y^3 ≤ (x + y)^3 = 8000
Therefore, the minimum possible value of the sum of the cubes of two positive real numbers whose sum is 20 is 8000.