#### Answer

Largest volume of a box is $=11664 in^3$ or, 115664 cubic inches with dimensions: $x=18,y=36$.
or
Largest volume of a box is $=11664 in^3$ or, 115664 cubic inches with dimensions: $36 \times 18 \times 18$.

#### Work Step by Step

Need to apply Lagrange Multipliers Method to determine the dimensions of a rectangular box of maximum volume.
we have $\nabla f=\lambda \nabla g$
Formula to calculate the maximum dimension is $4x+y=108$
Formula to find the volume of a box is $V=x^2y$
Simplify to get the value of $x,y$ and $z$
$\dfrac{dV}{dx}=12x(18-x)$
This gives, $x=0, y=108$; $V(0)=0$
$x=18, y=36$; $V(18)=11664$
$x=27, y=0$; $V(27)=0$
Apply Second derivative test: Some noteworthy points to calculate the local minimum, local maximum and saddle point of $f$.
1. If $D(p,q)=f_{xx}(p,q)f_{yy}(p,q)-[f_{xy}(p,q)]^2 \gt 0$ and $f_{xx}(p,q)\gt 0$ , then $f(p,q)$ is a local minimum.
2.If $D(p,q)=f_{xx}(p,q)f_{yy}(p,q)-[f_{xy}(p,q)]^2 \gt 0$ and $f_{xx}(p,q)\lt 0$ , then $f(p,q)$ is a local maximum.
3. If $D(p,q)=f_{xx}(p,q)f_{yy}(p,q)-[f_{xy}(p,q)]^2 \lt 0$ , then $f(p,q)$ is a not a local minimum and local maximum or, a saddle point.
Thus, critical points are: $(18,36)$
Therefore, the Largest volume of a box is $=11664 in^3$ or, 115664 cubic inches with dimensions: $x=18,y=36$.
or
Largest volume of a box is $=11664 in^3$ or, 115664 cubic inches with dimensions: $36 \times 18 \times 18$.