Question

数学高手进,a+b+c=1,ab+ac+bc=8/27,求abc的最大值和最小值.

Answer 1

令u=a+b ,v=ab,1>u>0

c=1-u

ab+ac+bc=v+u(1-u)=8/27

v=8/27-u+u^2

体积f(u)=abc=v(1-u)=(8/27-u+u^2)(1-u)=-u^3+2u^2-35u/27+8/27

f'(u)=-3u^2+4u-35/27=0

解得u1=5/9,u2=7/9

代人得最大值是20/729,最小值是16/729