But I don't think your last condition is enough. If f''' is strictly positive everywhere, then af''' is, too, and inf f will always be bigger than inf f-af'''.

Thank you for your effort! In the first example f-f''' takes negative values somewhere around -0.7 and 0.5 (see wolframalpha.com/input/?i=e%5E(−1%2Fx%5E2)%2F(x%5E10)(−8x%2B‌​36x%5E3−24x%5E5%2Bx%‌​5E10)) but it should be fine if one chooses a small enough. In the second example I don't know why you get this term, see wolframalpha.com/input/?i=d%5E3%2Fdx%5E3+exp(-1%2F(1%2Bx%5E2‌​)). Anyway, I found f(x) = -(x+1)^2 *exp(-x^2) which has minimum approx. -1,786856 whereas f-0.001*f''' has minimum approx. -1,785314 > inf f.