Let $f(x)$ be a polynomial with only negative non-leading coefficients ($a_n = 1$ since I only care about roots, so multiplicative constants are unimportant): $$ f(x) = x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0 \quad \forall i: a_i \le 0 $$ For a problem I'm working on I must be able to guarantee that there is always at least (ideally exactly) one positive real solution. Can I assume this? I tried to search for it on the internet but I don't know the proper terminology, so I didn't find anything.
I only was able to proof that if all roots are real that then at least one positive root must exist, because having only negative roots would cause all coefficients to be positive, which can't be (just foil $f(x) = (x + r_1) (x + r_2) \dots$ to see why).
I played around a bit with Wolfram|Alpha and it seems to be true, but this is obviously not a prove.