Lagrange Multipliers — Question 4

Question 4 – Solution

Strategy. The multiplier equations force equal positive side lengths; then compare with the zero-volume boundary and verify by AM–GM.

Step 1: Lagrange system For $g=x+y+z$, $$\nabla V=\left\langle yz,xz,xy\right\rangle =\lambda\left\langle 1,1,1\right\rangle.$$ Because $x,y,z>0$, equality of the three components gives $$yz=xz=xy,$$ and division by positive factors yields $x=y=z$. The constraint then gives $$\boxed{x=y=z=4}.$$

Step 2: Volume $$\boxed{V_{\max}=4\cdot 4\cdot 4=64}.$$

Step 3: Global verification Allowing $x,y,z\ge 0$ closes the feasible simplex. Its boundary has at least one zero side, so $V=0$ there. In the positive interior, AM–GM gives $$\frac{x+y+z}{3}\ge(xyz)^{1/3}.$$ Since the left side is $4$, $$xyz\le 64,$$ with equality only when $x=y=z$. Thus the Lagrange candidate is the unique positive absolute maximizer.