Help finding max value of function

Find the maximum value of f(x,y,z)=xy+xz+yz−4xyz subject to the constraints x+y+z=8 and x,y,z ≥ 0.

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We want to maximize f(x,y,z)=xy+xz+yz−4xyz subject to the g(x,y,z) = x + y + z = 8
(with x, y, z non-negative).

By Lagrange Multipliers, ∇f = λ∇g
==> = λ<1, 1, 1>.
==> λ = y + z - 4yz = x + z - 4xz = x + y - 4xy
==> y + z - 4yz = x + z - 4xz, and x + z - 4xz = x + y - 4xy
==> y(1 - 4z) = x(1 - 4z), and z(1 - 4x) = y(1 - 4x).

The first equation implies that y = x or z = 1/4.
(i) If y = x, then z(1 - 4x) = x(1 - 4x)
==> (z - x)(1 - 4x) = 0
==> x = 1/4 or x = z.

By using the constraint equation g,
x = 1/4, y = 1/4 ==> z = 15/2
x = y = z ==> x = y = z = 8/3.

(ii) If z = 1/4, then (1/4)(1 - 4x) = y(1 - 4x)
==> y = 1/4 or x = 1/4.

By using the constraint equation g,
z = 1/4, y = 1/4 ==> x = 15/2
z = 1/4, x = 1/4 ==> y = 15/2.
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Substitute these points into f:
f(1/4, 1/4, 15/2) = 31/16 (and also for (1/4, 15/2, 1/4) and (15/2, 1/4, 1/4))
f(8/3, 8/3, 8/3) = -1472/27.

Hence, the maximum value is 31/16.

I hope this helps!