Question

Tìm GTNN: A=|x+1|+|x+2|+|+x+3|+...+|x+2016|+100

Answer 1

A = |x + 1| + | x + 2| + |x + 3| + ............... + |x + 2016| + 100

Đặt : A' = |x + 1| + | x + 2| + |x + 3| + ............... + |x + 2016|

=> A' = |x + 1| + |-x - 2| + |x + 3| + ............... + |-x - 2016|

Áp dụng BĐT |a| + |b| \(\ge\) |a + b| , có :

|x + 1| + |-x - 2| + |x + 3| + ............... + |-x - 2016| \(\ge\) |x + 1 - x - 2 + x + 3 - x - 4 + ....... + x + 2015 - x - 2016|

<=> A' \(\ge\) |1 - 2 + 3 - 4 + ......... + 2015 - 2016| = |-1008| = 1008

=> A \(\ge\) 1008 + 100 = 1108

=> Min_A = 1108