Do you usually use the long procedures given in the standard textbooks in calculus or have you tried using some derived formulas like the one given in the link below? Perhaps, you can also try deriving other formulas for solving some other optimization problems and if you have already, may be you can also share your formulas with us.
Article On Minimum Distance Problem
Softools like matlab, mathematica etc or some FOSS may be used for the same.
Greetings, Romer!
I can only add to all previous recommendationыs, that if you using c++ for calculations, then open access library will be good for you
http://ab-initio.mit.edu/wiki/index.php/NLopt_C-plus-plus_Reference
Dear Romer,
The problem was considered (as a puzzle) even by Fibonacci in his Liber Abaci.
Two birds start flying from the tops of two towers 50 feet apart; one tower is 30 feet high and the other 40 feet high. Starting at the same time and flying at the same rate, the birds reach a fountain between the bases of the towers at the same moment. How far is the fountain from each tower?
Greetings, Tadeusz
You can see here a method that significantly simplifies the calculation of extremes in fractional functions:
Article EXTREMOS RELATIVOS DE FUNCIONES FRACCIONARIAS.
Dear Romer,
Minimum length problems show up everywhere in applications because adding length requires some energy and energy is one of those wonderful conservation principles. They are also important because length defines the geometry of the object under investigation and that has nice connection to topology, differential geometry (e.g., general relativity), etc. There is insight to be learned from deriving relations which might be specific to your application, and thus, not in any textbook. Many times you might also run into situations where there is no close form solution, and in which case, you’ll define a relationship as to move in a beneficial direction that gets you closer to your minimum (and that opens the door numerical methods). Hope this helps,
Alfonso
These links may be helpful for you
https://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/maxmindirectory/MaxMin.html
http://www.mathalino.com/reviewer/differential-calculus/43-45-solved-problems-in-maxima-and-minima
https://www.khanacademy.org/math/differential-calculus/derivative-applications
Dear Romer,
One of these methods is described in the paper "A strong separation theorem in normed linear spaces" by Octav Olteanu, published in Mathematica (Cluj), 35(58), 1 (1993), pp. 59 - 63. The method works in arbitrary real Hilbert space, and for a large class of closed convex (in general unbounded) subsets, with smooth boundaries. The general geometric idea is to prove firstly that in any (real) normed vector space, two convex subsets A, B with the distance d(A,B)>0, can be separated by closed parallel hyperplanes H_1, H_2, such that d(H_1, H_2) = d(A,B). In case of a Hilbert space, assuming that the distance d(A, B) is attained, the two mentioned above hyperplanes are orthogonal to the common normal for the subsets A,B, having smooth boundaries. Precisely, in this particular case, they are tangent hyperplanes to the surfaces defined by the boundaries of A, B. In particular, the method works in R^n and does not use the Lagrange's function defined by means of Lagrange's multipliers. Unfortunately, there is no electronic copy of this earlier paper.
Sincerely, Octav
In the particular case when the distance is attained, mentioned above, the hyperplanes H_1, H_2 are tangent respectively to the boundaries of A, B at the points where the distance d(A,B) is attained.
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