Áp dụng BĐT Bunhiacopski(mở rộng):
(4√a3+4√b3+4√c3)4
=(4√ax.4√ax.4√ax.4√x3+4√by.4√by.4√by.4√y3+
+4√cz.4√cz.4√cz.4√z3)4
≤(ax+by+cz)3(x3+y3+z3)=S
⇒S≥(4√a3+4√b3+4√c3)4
Dấu "=" xảy ra ⇔{axx3=byy3=czz3ax+by+cz=1⇔{ax4=by4=cz4ax+by+cz=1
⇔{x4√a=y4√b=z4√cax+by+cz=1
⇒ax+bx4√ba+cx4√ca=1 ⇒x=4√a(4√a3+4√b3+4√c3)
Tương tự :
{y=4√b(4√a3+4√b3+4√c3)z=4√c(4√a3+4√b3+4√c3)
Vậy: Min(S)=(4√a3+4√b3+4√c3)4