The domain of the function `f(x)=(log)_e{(log)_(|sinx|)(x^2-8x+23)-{3/((log)_2|sinx|)}}` contains which of the following interval(s)? (a)`(3,pi)` (b) `(pi,(3pi)/2)` (c) `((3pi)/2,5)` (d) none of these

Options

A. `(3, pi)`
B. `(pi,(3pi)/(2))`
C. `((3pi)/(2),5)`
D. None of these

Answer

A, B, C

Explanation

`f(x)` is defined if `log_(|sinx|)(x^(2)-8x+23)-(3)/(log_(2)|sinx|) gt 0`
`or log_(|sinx|)((x^(2)-8x+23)/(8)) gt 0`
This is true if `|sinx| ne 0,1 and (x^(2)-8x+23)/(8) lt 1.`
Now, `(x^(2)-8x+23)/(8) lt 1 or x^(2)-8x+15 lt 0`
`or x in (3,5) -{pi,(3pi)/(2)}`
Domain `=(3,pi) cup (pi,(3pi)/(2)) cup ((3pi)/(2),5).`