絕對值式子的最小值

1. 令 y = |x-a| + |x-b| + |x-c|當 x≤a 時, y = a-x+b-x+c-x = (a+b+c)-3x即y隨x增加而減小.當 a<x≤b 時,y = x-a+b-x+c-x = (-a+b+c)-x即y隨x增加而減小.當 b<x≤c 時,y = x-a+x-b+c-x = (-a-b+c)+x即y隨x增加而增加.當 c<x 時,y = x-a+x-b+x-c = -(a+b+c)+3x即y隨x增加而增加.所以y的最小值處於x = b時,而其時y = (-a+b+c)-b = c-a 2.令 y = |x-a| + |x-b| + |x-c| + |x-d|當 x≤a 時,y = a-x+b-x+c-x+d-x = (a+b+c+d)-4x即y隨x增加而減小.當 a<x≤b 時,y = x-a+b-x+c-x+d-x = (-a+b+c+d)-2x即y隨x增加而減小.當 b<x≤c 時,y = x-a+x-b+c-x+d-x = (-a-b+c+d)即y為常數.當 c<x≤d 時,y = x-a+x-b+x-c+d-x = (-a-b-c+d)+2x即y隨x增加而增加.當 d<x 時,y = x-a+x-b+x-c+x-d = -(a+b+c+d)+4x即y隨x增加而增加.明顯當 b≤x≤c 時y = -a-b+c+d為最小值