Question

va rog urgent

sa se demonstreze ca daca a,b,c sunt nr. reale astfel incat a+b+c≥3 atunci a²+b²+c²≥3

Answer 1

a+b+c≥3=>
(a+b+c)≥3³
a²+b²+c²+2ab+2bc+2ca≥9   relatia   1
dar   
a²+b²≥2ab
b²+c²≥2bc
a²+c²≥2ac
-------------------   se   aduna
2a²+2b²+2c²≥2ab=2bc+2ca
asadar
a²+b²+c²+(2a²+2b²+2c²)≥a²+b²+c²+2ab+2bc+2ca≥9=.
3a²+3b²+3c²≥9   
3*(a²+b²+c²)≥9   imparti   egalitatea    prin   3
a²+b²+c²≥3