Question

Ajuuutttooorrr repede va rog frumos! Am un exercitiu pe care nu stiu så il rezolv. Multumesc frumos!
1+a^2/b+1+b^2/a mai mare sau egal cu 4.

Answer 1

(1+a²)/b+(1+b²)/a≥4;   a si b≠0

stim ca 1-2a+a²≥0;   (1-a)²≥0

=>1+a²≥2a

la fel si 1+b²≥2b

=> (1+a²)/b+(1+b²)/a≥2a/b+2b/a

(1+a²)/b+(1+b²)/a≥(2a²+2b²)/ab

(1+a²)/b+(1+b²)/a≥2(a²+b²)/ab

dar a²+b²≥2ab,     (a²-2ab+b²≥0)

(1+a²)/b+(1+b²)/a≥2(a²+b²)/ab≥2·2ab/ab

=> (1+a²)/b+(1+b²)/a≥4