We want to maximize [tex]V(x,y,z)=xyz[/tex] subject to the constraint [tex]x+5y+4z=15[/tex].
The Lagrangian is
[tex]L(x,y,z,\lambda)=xyz+\lambda(x+5y+4z-15)[/tex]
with critical points where the partial derivatives vanish:
[tex]L_x=yz+\lambda=0\implies\lambda=-yz[/tex]
[tex]L_y=xz+5\lambda=0\implies\lambda=-\dfrac{xz}5[/tex]
[tex]L_z=xy+4\lambda=0\implies\lambda=-\dfrac{xy}4[/tex]
[tex]L_\lambda=x+5y+4z-15=0[/tex]
We assume [tex]\lambda\neq0[/tex]. Then from the first three equations, we get
[tex]-yz=-\dfrac{xz}5\implies x=5y[/tex]
[tex]-yz=\dfrac{-xy}4\implies x=4z[/tex]
[tex]-\dfrac{xz}5=-\dfrac{xy}4\implies5y=4z[/tex]
which leads us to a critical point at [tex](x,y,z)=\left(5,1,\frac54\right)[/tex], and hence a maximum volume of [tex]\frac{25}4[/tex].
