Question

4(a^6+b^6) >sau egal (a^2+ b^2)^3

Answer 1

[tex]4*(a^{6}+b^{6}) \geq (a^2+b^2)^3 \\ 4a^6+4b^6 \geq a^6+b^6+3a^4b^2+3a^2b^4 \\ 3a^6+3b^6-3a^4b^2-3a^2b^4 \geq 0 \\ a^6+b^6-a^4b^2-a^2b^4 \geq 0 \\ a^4(a^2-b^2)+b^4(b^2-a^2) \geq 0 \\ (a^4-b^4)(a^2-b^2) \geq 0 \\ (a^2-b^2)^2(a^2+b^2) \geq 0 \\ (a^2-b^2)^2 \geq 0~ (Adevarat!) \\ (a^2+b^2) \geq 0~(suma~de~patrate~mai~mari~ca~0) \\ Deci~inegalitatea~este~aratata![/tex]