Question

已知a大于0,b大于0,且a+b+c=1.求证1/a+1/b+1/c大于等于9

Answer 1

∵b/a+a/b≥2（√b/a×√a/b）=2×1=2

c/a+a/c≥2（√c/a×√a/c）=2×1=2

c/b+b/c≥2（√c/b×√b/c）=2×1=2

∴1/a+1/b+1/c=(a+b+c)/a+(a+b+c)/b+(a+b+c)/c

=1+b/a+c/a+a/b+c/b+1+a/c+b/c+1

=3+(b/a+a/b)+(c/a+a/c)+(c/b+b/c)

≥3+2+2+2

=9