area of a box is simply the sum of the areas of each of the sides so the constraint is given by2xy+ 2yz+ 2xz= 64=⇒xy+yz+xz= 32.Note that we divide the constraint by 2 to simplify the equation a little. Also, we get the functiong(x, y, z)from this.g(x, y, z) =xy+yz+xz.Here are the four equations that we need to solve.yz=λ(y+z)(fx=λgx),(3.4)xz=λ(x+z)(fy=λgy),(3.5)xy=λ(x+y)(fz=λgz),(3.6)xy+yz+xz= 32(g(x, y, z) = 32).(3.7)Although the equations are nonlinear, there are many ways to solve this system.We will solve it in thefollowing way. Let us multiply equation (3.4) byx,equation (3.5) byyandequation (3.6) byz.xyz=λx(y+z),(3.8)xyz=λy(x+z),(3.9)xyz=λz(x+y).(3.10)Now notice that we can set equations (3.8) and (3.9) equal. Doing this givesλx(y+z)=λy(x+z),λ(xy+xz)-λ(xy+yz)=0,λ(xz-yz)=0=⇒λ= 0orxz=yz.This implies two possibilities. The first,λ= 0,is not possible since if this is the caseequation (3.4) willreduce toyz= 0=⇒y= 0orz= 0.Since we are talking about the dimensions of a box neither of these are possible so we can discountλ= 0.This leaves the second possibility.xz=yz.109(MATH100)notes100-ch3.pdf downloaded by sclaw from at 2014-09-10 05:46:10. Academic use within HKUST only.