MATH2010 Multivariable Calculus and Matrix Algebra
Finding Lagrange Multipliers

Find the minimum amount of cardboard required to make an open topped box of volume 4m³.

The objective function and the constraint are given by

f(x,y,z) = xy + 2xz + 2yz (1)

g(x,y,z) = xyz = 4m³ (2)

We want to solve $\nabla f = \lambda \nabla g$. We have

$$\nabla f = \left[ \begin{array}{c} y+2z \\ x+2z \\ 2x+2y \e... ...g = \left[ \begin{array}{c} yz \\ xz \\ xy \end{array} \right]$$

so we need to solve the equations

    

$$\lambda yz$$ y+2z = (3)

$$\lambda xz$$ x+2z = (4)

$$\lambda xy$$ 2x+2y = (5)

xyz = 4m³ (6)

so we have four nonlinear equations in four variables.

One way to solve these equations is as follows:

• Take $y(\ref{yeqn})-z(\ref{zeqn})$:
  
  $$xy+2zy-2xz-2yz=0 \; \Rightarrow \; xy-2xz=0 \; \Rightarrow \; x=0 \mbox{ or } y=2z$$
  
  
  If x=0 then we violate equation (6). So we must have
  
   

  y=2z (7)

  
  

• Take $x(\ref{xeqn})-z(\ref{zeqn})$:
  
  $$xy+2zx-2xz-2yz=0 \; \Rightarrow \; xy-2yz=0 \; \Rightarrow \; y=0 \mbox{ or } x=2z$$
  
  
  If y=0 then we violate equation (6). So we must have
  
   

  x=2z (8)

  
  

• Substitute in equation (6) for x and y from equations (8) and (7):
  
  $$4z^3 = 4m^3 \; \Rightarrow \; z=m$$
  
  

• Equations (8) and (7) then give the final solution:
  
  $$(x,y,z)=(2m,2m,m), \mbox{ with value } f(2m,2m,m) = 12m^2$$