# Thread: Minimum sum equals sum of minima?

1. ## Minimum sum equals sum of minima?

Hello everyone!

If we have functions, f1(t), f2(t), etc. all positive in our observation window, plus:
$\displaystyle f_1(t) > f_2(t) > ... > f_N(t)$ for all $\displaystyle a<t<b$, does it follow that:
$\displaystyle \int ^b _a f_1(t)dt > \int ^b _a f_2(t)dt > ... > \int ^b _a f_N(t)dt$?

i.e. is the saying "minimum sum equals sum of minima" correct here?

Thanks!

2. ## Re: Minimum sum equals sum of minima?

It does follow from a basic theorem in analysis. That is, the theorem holds when N = 2, and I think you could induct on N.