0929132人目の素数さん
2018/11/25(日) 00:37:10.99ID:AuW29Ma5f '(x) < (3/2)^(1/3) f(x) = 1.14471424 f(x),
60th Putnam (1999/Dec/04) B-4
〔補題〕
lim(x→-∞) F(x) ≧0,
F '(x) > 0 for all x∈R
ならば
F(x) > 0 for all x∈R
(背理法で示せる。)
g(x) = (3/2)f(x)^3 - {f '(x)}^3,
とおくと
g '(x) = 3f '(x) {(3/2)f(x)^2 - f '(x)f "(x)} ≡ 3f '(x) h(x),
h '(x) = 3f(x) f '(x) - f '(x) f '''(x) - {f "(x)}^2
= f '(x) {f(x) - f '''(x)} + {2f(x) f '(x) - [f "(x)]^2}
≡ f '(x) {f(x) - f '''(x)} + L(x),
L '(x) = 2f '(x){f(x) - f '''(x)} + {f '(x)}^2 > 0,
補題により
L(x) = 2f(x) f '(x) - [f "(x)]^2 > 0,
h '(x) > 0,
補題により
h(x) = (3/2)f(x)^2 - f '(x)f "(x) > 0,
g '(x) > 0,
補題により
g(x) = (3/2)f(x)^3 - {f '(x)}^3 > 0,
f '(x) < (3/2)^(1/3) f(x),