Answer:
We can use the method of Lagrange multipliers to solve this problem. We need to minimize the objective function 4a * 5b * 6c subject to the given constraints a * b >= 11, a - b = 35 - 12b, a >= 0, b >= 0, c >= 0.
Let's define the Lagrangian function as follows:
L(a, b, c, λ1, λ2, λ3) = 4a * 5b * 6c + λ1(a * b - 11) + λ2(a - b - 35 + 12b) + λ3a + λ4b + λ5c
where λ1, λ2, λ3, λ4, λ5 are the Lagrange multipliers.
Now we need to find the critical points of L by taking partial derivatives of L with respect to a, b, c, λ1, λ2, λ3, λ4, λ5 and setting them equal to zero:
∂L/∂a = 20bcλ1 + λ2 + λ3 = 0
∂L/∂b = 12acλ1 - λ2 + λ4 = 0
∂L/∂c = 30abλ1 + λ5 = 0
∂L/∂λ1 = ab - 11 = 0
∂L/∂λ2 = a - b - 35 + 12b = 0
∂L/∂λ3 = a >= 0
∂L/∂λ4 = b >= 0
∂L/∂λ5 = c >= 0
Solving for a, b, and c yields:
a = 3√(55/4)
b = (3√(55/4) - 35)/11
c = 11/(2 * 3√(55/4))
Substituting these values back into the Lagrangian function L gives us the minimum value of the objective function:
L(3√(55/4), (3√(55/4) - 35)/11, 11/(2 * 3√(55/4)), λ1, λ2, λ3, λ4, λ5) = 77√(55)
Therefore, the minimum value of the expression 4a * 5b * 6c subject to the given constraints is 77√(55).