Question

已知a、b、c、d为正实数,a>b、c>d,若b/a<d/c,求证:a/b<b+d/a+c<d/c

Answer 1

估计原题是要证明：
a/b < (b+d)/(a+c) < d/c

b/a < d/c
两边同乘以ac (a > 0, c > 0, ac > 0):
bc < ad
(b+d)/(a+c) -b/a
= [a(b+d) -b(a+c)]/[a(a+c)]
=(ab + ad - ab - bc)/[a(a+c)]
=(ad - bc) /[a(a+c)]
已知bc < ad; a > 0, c > 0, a + c > 0
故(ad - bc) /[a(a+c)] > 0
(b+d)/(a+c) -b/a > 0
(b+d)/(a+c) > b/a

d/c - (b+d)/(a+c)
=[d(a+c) -c(b+d)]/[c(a+c)]
=(ad + cd -bc -cd)/[c(a+c)]
=(ad -bc)/[c(a+c)]
已知bc < ad; a > 0, c > 0, a + c > 0
故(ad -bc)/[c(a+c)] > 0
d/c - (b+d)/(a+c) > 0
(b+d)/(a+c) < d/c

Answer 2

a/b<b+d/a+c<d/c ?
a/b < b+ d/a + c < d/c ?
还是a/b < (b+ d)/(a + c) < d/c 啊