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If alpha < beta < gamma and sin gamma co...

If `alpha < beta < gamma` and `sin gamma cos alpha=1,` where `alpha,gamma in[pi,2 pi],` then the least integral value of `f(x) = | x - alpha| + | x - beta| + |x - gamma|` is

A

0

B

1

C

2

D

3

Text Solution

Verified by Experts

The correct Answer is:
C
`sin gamma.cos alpha =1 alpha, gamma in [pi, 2pi]`
`therefore sin gamma = cos alpha =1`
`rArr gamma = pi//2, alpha = 2pi` (rejected) `(as alpha lt beta lt gamma)`
Other possibility is `sin gamma = cos alpha =-1 rArr gamma = 3 pi//2, alpha = pi`
`f(x)|_(min)=f(beta)=beta-alpha+0+gamma-beta`
`=gamma -alpha`
`=(3pi)/(2)-pi=(pi)/(2)`
`f(x)ge pi//2 rArr` least integral value of f(x) is 2.
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