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 ?

wolframalpha says that the maximum is at (1/3,1/3,1/3)

So, f(x,y,z) is 1/243

wrong there is other max but wolfram dpoesnt give excct val

To find the maximum value of x^3*y^3 + y^3*z^3 + z^3*x^3, we can apply the AM-GM inequality.

The AM-GM inequality states that for a set of non-negative real numbers, the arithmetic mean (AM) is always greater than or equal to the geometric mean (GM). Mathematically, it can be written as:

AM ≥ GM

Now, applying this inequality to the expression x^3*y^3 + y^3*z^3 + z^3*x^3, we have:

(x^3*y^3 + y^3*z^3 + z^3*x^3)/3 ≥ (x^3*y^3 * y^3*z^3 * z^3*x^3)^(1/3)

Simplifying this, we get:

(x^3*y^3 + y^3*z^3 + z^3*x^3) ≥ 3(x^3*y^3 * y^3*z^3 * z^3*x^3)^(1/3)

Next, we can substitute the given condition x + y + z = 1 into the expression. We can rewrite it as:

(x^3*y^3 + y^3*z^3 + z^3*x^3) ≥ 3(xyz)^2

Now, since x, y, and z are non-negative real numbers and x + y + z = 1, the maximum value of xyz occurs when x = y = z = 1/3. Substituting these values into the inequality, we have:

(1/27 + 1/27 + 1/27) ≥ 3(1/27)^2

1/9 ≥ 3/729

Multiplying by 729/3 on both sides, we get:

81 ≥ 1

Since this is true, we know that the maximum value of x^3*y^3 + y^3*z^3 + z^3*x^3 is 1/9.

Therefore, the value of a/b is 1/9, where a = 1 and b = 9. The sum of a and b is 1 + 9 = 10.

So, the value of a+b is 10.