This box volume is described as
maxV=xyz
subject to
x - s_1^2= 0
y-s_2^2= 0
z-s_3^2 = 0
x + 8y + 7z = 24
We introduce the slack variables s_1,s_2,s_3 to transform the inequality into equality restrictions.
The lagrangian is
L(X,S,Lambda)=xyz+lambda_1(x-s_1^2)+lambda_2(y-s_2^2)+lambda_3(z-s_3^2)+lambda_4(x+8y+7z-24)
Here
X=(x,y,z)
S=(s_1,s_2,s_3)
Lambda=(lambda_1,lambda_2,lambda_3,lambda_4)
Now the stationary points are given by the solutions of
grad L(X,S,Lambda)= vec 0
{(lambda_1 + lambda_4 + y z=0), (lambda_2 + 8 lambda_4 + x z=0),
(lambda_3 + 7 lambda_4 + x y=0), (-2 lambda_1 s_1=0), (-2 lambda_2 s_2=0), (-2 lambda_3 s_3=0), (-s_1^2 + x=0), (-s_2^2 +
y=0), (-s_3^2 + z=0), (-24 + x + 8 y + 7 z=0):}
Solving for (X,S,Lambda) we have a meaningful sample
(x = 8, y = 1, z = 8/7, s_1 = -2 sqrt[2], s_2= -1,
s_3= 2 sqrt[2/7], lambda_1= 0, lambda_2= 0, lambda_3= 0, lambda_4 = -8/7)
we rejected stationary points with x=0 or y=0 or z=0
which implied on null volumes.
The shown solution has s_1 ne 0, s_2 ne0, s_3 ne 0 so is an interior solution.
The Lambda found lambda_1=lambda_2=lambda_3=0, lambda_4 ne 0 show the actuating and non actuacting restrictions. The lambda_k = 0 for non actuating restrictions and lambda_4 ne 0 for the actuating one.
The found volume is
V = 64/7 volume units.
