ПОВТОРЕННЯ НАВЧАЛЬНОГО МАТЕРІАЛУ » §.1 (8.4-6)

4) (a + b + c)2 ≤ 3(a2 + b2 + c2). a2 + b2 + c2 + 2ab + 2bc + 2ac – 3a2 – 3b2 – 3c2 ≤ 0. 2a2 + 2b2 + 2c2 – 2ab – 2bc – 2ac ≥ 0. a2 + a2 + b2 + b2 + c2 + c2 – 2ab – 2bc – 2ac ≥ 0. (a – b)2 + (a – c)2 + (b – c)2 ≥ 0. Доведено. 5) 3a/b + b/27a ≥ 2/3; a > 0, b > 0. (81a^2+ b^2- 19ab)/27ab ≥ 0. (b^2- 2•b •9a+(9a)^2 )/27ab ≥ 0. ((b-9a)^2)/27ab ≥ 0, бо (b – 9a)2 ≥ 0. 27ab > 0. 6) a3 + 8 ≥ 2a2 + 4a; a ≥ –2. a3 – 2a2 – 4a + 8 ≥ 0. a2(a – 2) – 4(a – 2) ≥ 0. (a – 2)(a2 – 4) ≥ 0. (a – 2)2(a + 2) ≥ 0. (a – 2)2 ≥ 0 i a + 2 ≥ 0, бо а ≥ –2 за умовою.