Step-by-step explanation:
By using the Divergence Theorem, the flux equals
∫∫∫ div F dV
= ∫∫∫ ((-2x - 4y) + (-6z) + 12) dV
= ∫(z = 0 to c) ∫(y = 0 to b) ∫(x = 0 to a) (12 - 2x - 4y - 6z) dz dy dx
= ∫(z = 0 to c) ∫(y = 0 to b) (12a - a^2 - 4ay - 6az) dy dx
= ∫(z = 0 to c) (12ab - a^2 b - 2ab^2 - 6abz) dx
= 12abc - a^2 bc - 2ab^2 c - 3abc^2.
So, we need to find the maximum value of
f(a,b,c) = 12abc - a^2 bc - 2ab^2 c - 3abc^2, with a,b,c > 0.
First, we find the critical points of f.
f_a = 12bc - 2abc - 2b^2 c - 3bc^2 = bc(12 - 2a - 2b - 3c)
f_b = 12ac - a^2 c - 4abc - 3ac^2 = ac(12 - a - 4b - 3c)
f_c = 12ab - a^2 b - 2ab^2 - 6abc = ab (12 - a - 2b - 6c).
Setting these equal to 0 (and remembering that a, b, c > 0):
12 - 2a - 2b - 3c = 0
12 - a - 4b - 3c = 0
12 - a - 2b - 6c = 0
Solving this system yields a = 3, b = 3/2, c = 1.
By the Second Derivative Test or otherwise, this can easily be checked to yield the maximum flux.
This maximal flux equals f(3, 3/2, 1) = 27/2
