x,y and z are positive reals. What is the maximum value of ((x+y+z)^3−x^3−y^3−z^3)^2/(x^2+y^2+z^2)^3−x^6−y^6−z^6
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To find the maximum value of the given expression, we need to determine the critical points where the derivative is zero. We will start by finding the derivative of the expression and then solving for when it equals zero.
Let's denote the given expression as f(x, y, z). We find the derivative of f(x, y, z) with respect to x, y, and z, respectively:
∂f/∂x = 2((x + y + z)^3 - x^3 - y^3 - z^3)((x^2 + y^2 + z^2)^3 - x^6 - y^6 - z^6)^(-1/3) * (3(x + y + z)^2 - 3x^2) - 6(x^2 + y^2 + z^2)^2 - 6x^5,
∂f/∂y = 2((x + y + z)^3 - x^3 - y^3 - z^3)((x^2 + y^2 + z^2)^3 - x^6 - y^6 - z^6)^(-1/3) * (3(x + y + z)^2 - 3y^2) - 6(x^2 + y^2 + z^2)^2 - 6*y^5,
∂f/∂z = 2((x + y + z)^3 - x^3 - y^3 - z^3)((x^2 + y^2 + z^2)^3 - x^6 - y^6 - z^6)^(-1/3) * (3(x + y + z)^2 - 3z^2) - 6(x^2 + y^2 + z^2)^2 - 6z^5.
To find the critical points, we set these partial derivatives equal to zero and solve for x, y, and z. However, the equations involved are quite complex for a human to solve manually. In this case, it may be more practical to use numerical methods or optimization software to find the critical points.
Once we find the critical points, we can evaluate the expression at each critical point as well as at the boundaries of the allowed domain to determine the maximum value of the given expression.
Please note that due to the complexity of the problem, it is highly recommended to use optimization algorithms or software tools to find the critical points and calculate the maximum value accurately.