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anemone
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If $s,\,t,\,u,\,v$ are positive integers with sum $63$, find the maximum value of $st+tu+uv$.
MarkFL said:My solution:
We are given the objective function:
\(\displaystyle f(s,t,u,v)=st+tu+uv\)
Subject to the constraint:
\(\displaystyle g(s,t,u,v)=s+t+u+v-63=0\)
Applying Lagrange multipliers, we obtain the system:
\(\displaystyle t=\lambda\)
\(\displaystyle s+u=\lambda\)
\(\displaystyle t+v=\lambda\)
\(\displaystyle u=\lambda\)
From this, we readily see $t=u$ and then $s=v=0$. Using the constraint, we then find:
\(\displaystyle 2t=63\implies t=u=\frac{63}{2}=31.5\)
Since we require the variables to be positive integers, let $(s,t,u,v)=(1,30,31,1)$ (or equivalently $(s,t,u,v)=(1,31,30,1)$) as these permutations are the closest to the real number maximum, then we find:
\(\displaystyle f_{\max}=f(1,30,31,1)=1\cdot30+30\cdot31+31\cdot1=30+930+31=991\)
anemone said:If $s,\,t,\,u,\,v$ are positive integers with sum $63$, find the maximum value of $st+tu+uv$.
kaliprasad said:$st+tu+uv = (s+u)(v+t) -vs$
as in the 1st term v and s do not lie in isolation we can maximize $(s+u)(v+t)$ and then minimize $vs$ independently . clearly $(s+u)(v+t)$ is maximum when they are as close as possible
so $s+ u = 32$ and $v+t = 31$ or $s+u = 31$ and $v+ t = 32$
and vs is minimum when $v=s = 1$
so we have $s= 1, u= 31, v= 1, t= 30$ or $s = 1, u = 30, t= 31, v= 1$ and in both cases $st+tu+uv = 31 * 32-1 = 991$ maximum
The maximum value of $st+tu+uv$ is 1323, which is achieved when each variable is equal to 21.
The maximum value of $st+tu+uv$ can be determined by finding the average of the sum, which in this case is 63 divided by 4, or 15.75. Then, assign each variable as close to that average as possible while still maintaining a sum of 63. For this problem, each variable should be equal to 21 to achieve the maximum value of 1323.
Yes, the maximum value of $st+tu+uv$ will change based on the sum of $s+t+u+v$. The key is to find the average of the sum and assign each variable as close to that average as possible while still maintaining the given sum.
Yes, there is a general formula for finding the maximum value of $st+tu+uv$ given any sum of $s+t+u+v$. The formula is: $max(st+tu+uv) = \frac{(s+t+u+v)^2}{4}$
Changing the values of $s,t,u,$ and $v$ will change the maximum value of $st+tu+uv$. As long as the sum of the variables remains the same, the maximum value will remain the same. However, changing the individual values will change the distribution of the sum and therefore change the maximum value of $st+tu+uv$.