The numerator and denominator are both homogeneous twice, so if g(x, y, z) is the numerator, r(x, y, z) is the denominator.
The value of f=g/r satisfies f(kx, ky, kz)=f(x, y, z), so it can be defined as r= 1 to find the maximum value.
This is equivalent to limiting r(x, y, z)= 1, and finding the maximum value of g(x, y, z) can start Lagrange.
The machine is broken.
More simply, note that r= 1 is the unit sphere S. In order to get the maximum value of G on S, the gradient of G is needed.
Grad(g) is perpendicular to S. grad(g) = (gx, gy, gz) = (y, x+2z, 2y).
The normal vector of S at (x, y, z) is (x, y, z).
Therefore, the above conditions are equivalent to
Y/x = (x+2z)/y = 2y/z = undetermined constant k
This is y = kx, z = 2x, and k * k = 5.
Thus, the radical number 5 with f = k/2 = 2 is obtained.
Note that this method is actually consistent with Lagrangian method.