The discussion revolves around finding the maximum value of the expression $st + tu + uv$ given that the positive integers $s, t, u, v$ sum to 63. The focus is on exploring different approaches to this optimization problem.
The discussion does not show clear consensus or disagreement, as the details of the proposed solutions are not fully articulated.
Limitations include the lack of detailed mathematical steps in the proposed solutions and the absence of specific assumptions or conditions that might affect the optimization.
MarkFL said:My solution:
We are given the objective function:
$$f(s,t,u,v)=st+tu+uv$$
Subject to the constraint:
$$g(s,t,u,v)=s+t+u+v-63=0$$
Applying Lagrange multipliers, we obtain the system:
$$t=\lambda$$
$$s+u=\lambda$$
$$t+v=\lambda$$
$$u=\lambda$$
From this, we readily see $t=u$ and then $s=v=0$. Using the constraint, we then find:
$$2t=63\implies t=u=\frac{63}{2}=31.5$$
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:
$$f_{\max}=f(1,30,31,1)=1\cdot30+30\cdot31+31\cdot1=30+930+31=991$$
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