>>522 (D1)

 f '(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),