You need to randomly select three positive numbers such that: x, y,z.
You know that the sum of the numbers is 100, hence:
`x + y+ z = 100 =gt 3x + 3 = 100 =gt 3x - 97 = 0`
Hence, the function that needs to be optimized is `f(x,y,z) = xyz ` and the constraint is `g(x,y,z)=x + y+ z = 100` .
Hence, you need to solve the equations:
`f_x = lambda*g_x =gt yz = lambda`
`f_y = lambda*g_y =gt xz = lambda`
`f_z = lambda*g_z =gt xy = lambda`
`x + y + z = 100`
You need to set the equations yz and xz equal such that:
`yz = xz`
You may divide by z since `z!=0` such that:
`x = y`
You need to set the equations yz and xy equal such that:
`yz = xy`
You may divide by y since `y!=0` such that:
`x = z`
Since x=z and x=y yields y=z, thus you may substitute x for y and z in constraint `x + y + z = 100` such that:
`x+x+x = 100 =gt 3x = 100 =gt x = 100/3 =gt x = 33.33`
Hence, evaluating the positive numbers for the product of these numbers to be maximum yields: `x=y=z~~33.33` .