# Rectangular box of largest surface area inscribed in a sphere of radius 1?

I am having trouble trying to follow a textbook example of such a problem. Using the Lagrange's Multiplier method, we could set up a set of equations like below:

$f(x,y,z)=8(xy+yz+zx)$

$g(x,y,z)=x^2+y^2+z^2-1=0, x>0, y>0, z>0$

then the lambda could be found by solving

$8(y+z,z+x,x+y)=2\lambda(x,y,z)$

suddenly, the textbook jumps right to the conclusion that

$8(x+y+z)=\lambda(x+y+z)$

$\lambda=8$

How is it derived exactly? I've tried googling and searching for answers but I had no luck finding anything. If someone could break it down step by step, it'd be greatly appreciated. Thanks.

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## 1 Answer

Just before the "jump" you have three equations $$\begin{align} 8(y+z)&=2\lambda x\\ 8(z+x)&=2\lambda y\\ 8(x+y)&=2\lambda z \end{align}$$ Add the three left hand sides and the three right hand sides and do a bit of dividing. What do you get?

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