Minimise sum of increasing functions with linear constraint

Question

Let $f(x)$ and $g(x)$ be two increasing continuous functions.

Given that $x_1 + x_2 = k$, show that the minimum of $f(x_1) + g(x_2)$ occurs where $f(x_1) = g(x_2)$

Answer 1

I'm not sure so check it please, but I think that the cost is evaluated by the integral of the two functions (in that case the area under the graph of the functions), hence if we must supply $1000$MW as constraint, the minimum of the area is when the two functions intersect each other; any other choice has a greater cost, for instance: if Area A supplies $800$MW and Area B $200$MW, the constraint is right but the cost (the integral) is equal to the minimum one plus the area between the two functions in the range $[700$MW$,800$MW$]$, so we would have a higher cost compared to the following choice: Area A supplies $700$MW and Area B $300$MW. I hope it helps.