Can AM-GM Inequality Solve This Algebraic Problem?

[tsimone75]

Homework Statement

{(x+y+z)^3-2(x+y+z)(x^2+y^2+z^2)}/xyz ≤ 9

Homework Equations

AM-GM inequality x+y+z ≥ 3(√xyz)(cube root) and xy+yz+zx ≥ 3√(xyz)^2(cube root)

The Attempt at a Solution

This is my attempt but I don't know if I am using the AM-GM inequality correctly

(x+y+z)^3/xyz ≤ 9+ [2(x+y+z)(x^2+y^2+z^2)]/xyz

(x+y+z)^3/xyz ≤ [9xyz+ 2(x+y+z){(x+y+z)(x+y+z)-2(xy+yz+zx)}]/xyz

(3√xyz)3/xyz ≤ [9xyz +2(x+y+z)^3 -4(x+y+z)(xy+yz+zx)]/xyz
27xyz/xyz ≤ [9xyz +2(3√xyz)^3 -4(3√xyz)(3√(xyz)^2)]/xyz

27xyz/xyz ≤ [9xyz +54xyz -36xyz]/xyz
27 ≤ 27

Is this correct?

 

[mighty2000]

I would like to provide some feedback on your solution. The AM-GM inequality states that for non-negative real numbers, the arithmetic mean is greater than or equal to the geometric mean. In other words, for any set of positive numbers, the sum of the numbers divided by the number of numbers is greater than or equal to the product of the numbers raised to the power of 1/number of numbers. In this case, we have three numbers (x, y, and z), so the inequality would be:

(x+y+z)/3 ≥ (xyz)^(1/3)

Using this inequality, we can rewrite the original expression as:

(x+y+z)^3/xyz ≤ 9 + 2(x+y+z)(x^2+y^2+z^2)/xyz

Applying the AM-GM inequality to the first term on the right side, we get:

(x+y+z)^3/xyz ≤ 9 + 2(x+y+z)(x^2+y^2+z^2)/xyz ≤ 9 + 2(3√xyz)(xy+yz+zx)/xyz

Using the AM-GM inequality again on the second term on the right side, we get:

(x+y+z)^3/xyz ≤ 9 + 2(3√xyz)(xy+yz+zx)/xyz ≤ 9 + 2(3√(xyz)^2)(x+y+z)/xyz

Simplifying, we get:

(x+y+z)^3/xyz ≤ 9 + 6(x+y+z)/xyz

Now, we can use the fact that (x+y+z)/3 ≥ (xyz)^(1/3) to rewrite the right side as:

(x+y+z)^3/xyz ≤ 9 + 2(xyz)^(1/3)

Finally, we can use the fact that (xyz)^(1/3) = (xyz)^2/3 to get:

(x+y+z)^3/xyz ≤ 9 + 2(xyz)^2/3

This is as far as we can simplify the expression using the AM-GM inequality. We can see that the expression on the right side is not equal to 9, so the original inequality is not true. This means that your attempt is not correct. However, your use of the AM-GM inequality is correct. Keep in mind that the AM-GM inequality does not always guarantee that the equality