Question

a) Demonstrati ca a² + b² + c² ≥ ab + bc + ac, ∀ a,b,c ∈ R
b) Aratati ca a^4 + b^4 + c^4 ≥ a²b² + b²c² + a²c², ∀ a,b,c ∈ R
c) Aratati ca a^4 + b^4 + c^4 ≥ abc(a + b + c), ∀ a,b,c ∈ R

Answer 1

a. a²+b²+c²≥ab+bc+ac ⇔
2a²+2b²+2c²-2ab-2bc-2ac≥0 ⇔
a²-2ab+b²+b²-2bc+c²+c²-2ac+a²≥0 ⇔
(a-b)²+(b-c)²+(a-c)²≥0 propozitie adevarata pentru orice a;b;c∈R;
b. a⁴+b⁴+c⁴≥a²b²+b²c²+a²c² ⇔
2a⁴+2b⁴+2c⁴-2a²b²-2b²c²-2a²c²≥0 ⇔
a⁴-2a²b²+b⁴+b⁴-2b²c²+c⁴+c⁴-2a²c²+a²≥0 ⇔
(a²-b²)²+(b²-c²)²+(c²-a²)²≥0 propozitie adevarata pentru orice a;b;c∈R;
c. a⁴+b⁴+c⁴≥abc(a+b+c)=a²bc+b²ac+c²ab propozitie adevarata pentru orice a;b;c∈R;