把8分为两个非负数之和,使其立方和最小,则分法是什么

Question

把8分为两个非负数之和,使其立方和最小,则分法是什么

Answer 1

a>=0;b>=0 ;a+b=8 => a+b>=2√ab =>8>=2√ab => 4>=√ab =>16>=ab => -16<=-ab
a^3+b^3=(a+b)(a^2-ab+b^2)=(a+b)[(a+b)^2-3ab]=8[64-3ab].(*)
-3ab>=-48 => 64-3ab>=64-48 =>64-3ab>=16 =>(*)>=8*16=128
=>min=128,此时a=b=4