Let x,y,z be non-negative real numbers satisfying the condition x+y+z=1. The maximum possible value of
x^3*y^3+y^3*z^3+z^3*x^3
has the form a/b where a and b are positive, coprime integers. What is the value of a+b?
To find the maximum value of x^3*y^3 + y^3*z^3 + z^3*x^3, we can use the AM-GM inequality and the given condition x + y + z = 1.
By the AM-GM inequality, we have:
x/3 + x/3 + x/3 ≥ 3√(x/3 * x/3 * x/3)
x/3 ≥ √(x^3/27)
x^3 ≤ 27/27
x^3 ≤ 1/3
Similarly, we have y^3 ≤ 1/3 and z^3 ≤ 1/3.
Now, let's determine when equality occurs in the AM-GM inequality. For equality to occur, all the terms inside the inequality must be equal. This happens when x/3 = y/3 = z/3, which implies x = y = z.
Since x + y + z = 1, we have x = y = z = 1/3.
Now, substituting these values in x^3*y^3 + y^3*z^3 + z^3*x^3, we get:
(1/3)^3 * (1/3)^3 + (1/3)^3 * (1/3)^3 + (1/3)^3 * (1/3)^3
= 1/3^6 + 1/3^6 + 1/3^6
= 3/3^6
= 1/3^5
So the maximum possible value of x^3*y^3 + y^3*z^3 + z^3*x^3 is 1/3^5.
The form of a/b where a and b are positive, coprime integers can be obtained by writing 1/3^5 as 1/243.
Therefore, a = 1 and b = 243, so the value of a + b is 1 + 243 = 244.