The number of integral solution of the equation `(sgn((x-1)(x-5)))^(x)=1` lying in the interval `[-10, 10]` is not equal to :

Options

A. `11`
B. `16`
C. `17`
D. `18`

Answer

A, B, C

Explanation

The number of ………..
`(f(x))^(g(x)) = 1` is possible when
Case - I `f(x)=1, g(x) in R`
Case - II `f(x) in R - {0}, g(x) = 0`
Case - III `f(x)= -1, g(x) =` even
Case - I `sgn((x-1)(x-5)) = 1`
`implies (x-1) (x-5) gt 0`
`implies x in (-oo, 1) uu (5, oo)`
Also `x in [-10, 10]`
`:. x = {-10, -9, -8, ..... -1,0}uu{6,7,8,9,10}`
`:. 16` values
Case - II
`g(x) = 0` and `f(x) != 0`
`x = 0` and `sgn{(x-1) (x-5)}!= 0`
i.e. already considered in Case - I
Case - III
`f(x)= -1, g(x) =` even
`implies sgn ((x-1) (x-5)) = -1` and `x =` even
`implies x in (1,5)` and `x =` even
`implies x = {2, 4} implies 2` more values
`:. 16+12=18`