Just a calculus problem:
As a function of $K \geq 1$, what is the minimum value of $f/a + f/b + f/c + f/d + f/e$ subject to the following constraints? $\begin{cases} 1 \leq a \leq c \\ 1 \leq b \leq c \\ 1 \leq d \\ 1 \leq e \\ f = \frac{a^2 b^2 d e}{c} \\ f = K \end{cases}$
I am fine with a reasonably detailed "here is how you do this calculus" answer, or "here is how to ask wolfram alpha/maple" answer (that works). I need to be able to handle variations (the formula for f is always a "monomial" but the powers on the variables can change, and the inequalities amongst the $a,b,c,d,e$ variables might change slightly, though all of them are always at least 1).
A version I can do: As a function of K ≥ 1, find the minimum value of $f/a + f/b + f/c + f/d + f/e$ subject to the following constraints: $\begin{cases} 1 \leq a,b,c,d,e \\ f = abcde \\ f = K \end{cases}$
This version is highly symmetric and I basically understand the region I am optimizing over. Setting any variable to 1 results in a highly non-optimal solution, so the minimum occurs in the middle of the surface where abcde = K, and so the gradient of the objective function is a scalar multiple of the normal to the surface. Both are very symmetric and the algebra involved in solving them is almost silly. The answer is the expected $a=b=c=d=e=K^{1/5}$ due to symmetry.
Motivation: In the background, $a,b,c,d,e,f$ are all positive odd integers describing the structure of an unknown group. In the previous incarnations of this problem, I assumed they were real numbers bounded below by 1, and the calculus minimum was in fact the group theory minimum.
On the new problem, I asked maple to give the unconstrained (except for the "f=" constraints) problem a shot, and it claims there is only one local extrema, and it involves a lot of negative numbers. I guess that means the minimum is at a "corner" (discontinuity of the function defining the boundary of the feasible region), but I have no idea what that means in more than 2 dimensions, and I am a little nervous that such an answer is wrong, at least from the group theory standpoint.