Question

若a,b,c为正数,求证a4+b4+c4≥abc(a+b+c) 注:4是的a,b,c的4次方

Answer 1

a^4+b^4+c^4=1/2(a^4+b^4+a^4+c^4+b^4+c^4)

≥1/2(2a^2*b^2+2a^2*c^2+2b^2*c^2)=a^2*b^2+a^2*c^2+b^2*c^2

=1/2(a^2*b^2+a^2*c^2+a^2*b^2+b^2*c^2+a^2*c^2+b^2*c^2)

=1/2(a^2*(b^2+c^2)+b^2*(a^2+c^2)+c^2*(a^2+b^2))

≥1/2(a^2*2bc+b^2*2ac+c^2*2ab)

=a^2bc+b^2*ac+c^2*ab

=abc(a+b+c)

对于任何实数都成立.