Maximising a Product of Numbers With Constant Sum

How do we maximise the product of three positive numbers subject to the sum being a constant?
Let the numbers be  
\[a, \; b, \; c\]
  so that  
\[a+b+c=K\]
  for some constant  
\[K\]
. We want to maximise  
\[P=abc\]
.
Rearrange  
\[a+b+c=K\]
  to give  
\[a=K-b-b\]
  then  
\[P=(K-b-c)bc=Kbc-b^2c-bc^2\]
.
For a maximum the partial derivatives with respect to  
\[b\]
  and  
\[c\]
  will be zero, so
\[\frac{\partial P}{\partial b}=Kc-2bc-c^2=c(K-2b-c)=0\]

\[\frac{\partial P}{\partial c}=Kb-b^2-2bc=b(K-b-2c)0\]

Obviously  
\[b, \; c \neq 0\]
  since then  
\[P=0\]
  so  
\[K-2b-c=K-b-2c=0\]
.
The solution is  
\[b=c= \frac{K}{3}\]
  so then  
\[a= \frac{K}{3}\]
  and the maximum value of  
\[P\]
  is  
\[\frac{K}{3} \times \frac{K}{3} \times \frac{K}{3} = \frac{K^3}{27}\]
.
We can use the same method for any number of variables.

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