Page 114 - MATINF Nr. 3

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114 PROBLEME DE MATEMATICA PENTRU CONCURSURI

M 98. Determinat , i cel mai mare num˘ar real k pentru care inegalitatea

2
2
2
(a + k)(b + k)(c + k) ≤ (1 + k) 3
are loc pentru orice a, b, c ∈ [0, ∞) astfel ˆıncˆat a + b + c = 3.

Leonard Giugiuc, Drobeta Turnu Severin s , i Costel B˘alc˘au, Pites , ti

M 99. Se consider˘a funct , ia f : [0, ∞) → [0, 1) astfel ˆıncˆat

» »
ln 1 + f(x) = x + ln 1 − f(x) , ∀x ∈ [0, ∞).

a) Demonstrat , i c˘a ecuat , ia funct , ional˘a are solut , ie.

b) Ar˘atat , i c˘a f admite primitive.
√
Z a b−1 Z b 1−a 1 + x
c) Fie a, b ∈ (0, 1) cu a < b. Ar˘atat , i c˘a f(x) dx + ln √ dx > 1.
1 − x
0 0
Floric˘a Anastase, Lehliu Gar˘a

M 100. Fie f, g : [a, b] → R dou˘a funct , ii derivabile cu derivatele continue astfel ˆıncˆat
2
2
f (x) + g (x) 6= 0, oricare ar fi x ∈ [a, b]. Demonstrat , i c˘a
s

b 0 0 2 2
(f (x)) + (g (x)) f (b) + g (b)
Z 2 2
dx ≥ ln .
2
2
2
2
f (x) + g (x) f (a) + g (a)
a
Cristinel Mortici, Viforˆata