Verify that the sum of three quantities x, y, and z, whose product is a constant k, is maximum when these three quantities are equal.

Answer 1

x y z = k

sum = s = x + y + z

ds = ds/dx dx + ds/dy dy + ds/dz dz
= 0 for max or min
but ds/dx = ds/dy = ds/dz = 1
so
ds = dx + dy + dz = 0
so max when dx = -dy - dz
but
x = k/(yz)
dx/dy = -kz /(yz)^2
dx/dz = -ky /(yz)^2
so
dx = -[-kz/((yz)^2 ]dy - [-ky/(yz)^2]dz
so k y = k z
so y = z

Answer 2

xyz=k therefore z=k/xy
S=x+y+k/xy
dS=(δS/δx)dx+(δS/δy)dy
=(1-k/x^2y)dx+(1-k/y^2x)dy
For min or max
1-k/x^2y=0,1-k/y^2x=0
Therefore k/x^2y=1 so x=k/xy
And k/y^2x=1 so y=k/xy
So x=y=z=k/xy