Question

已知正数a,b,c满足a+b+c=1,且1/a+1/b+1/c=10,求abc的最小值

Answer 1

1/a+1/b+1/c
=(1/a+1/b+1/c)*(a+b+c)
=(b+c)/a + (a+c)/b + (a+b)/c +3
=(b/a+a/b)+(c/b+b/c)+(a/c+c/a)+3
≥2√(b/a * a/b) + 2√(c/b * b/c) + 2√(a/c * c/a) +3
=9
最小值就是9