Consider the following minimization problem: $$ \text{minimize} \quad \frac{( a + b )^2}{y - x} +\frac{a^2}{x} +\frac{b^2}{1 - y} \quad \text{in} \quad 0 < x < y < 1, $$ where $a$ and $b$ are fixed positive numbers. Note how the minimization requirement forces 1) $x$ and $y$ to stay away from each other, 2) $x$ to stay away from $0$ and 3) $y$ to stay away from $1$.
This problem is quite easy to solve directly by taking derivatives and such, but in view of some possible generalizations which are of interest for me, I wonder whether there is a simple trick to find the answer. A trick which involves as little computations as possible.
I know my question is a bit vague, but I'd appreciate any ideas. I'd also like to hear about different possible approaches to similar minimization problems.