The number of integral values of x for which the function `sqrt(sinx+cosx)+sqrt(7x-x^(2)-6) ` is defined is ________.

Options

A. `[1, 3pi//4] uu [7pi//4, 6]`
B. `[1, 3pi//4]uu [ 6- pi//4, 6]`
C. `[1, 6]`
D. none of these

Answer

A

Explanation

We have,
`f(x) = sqrt(sin x + cos x) + sqrt(7x- x^(2) - 6)`
`rArr (x) = 2^(1//4) sqrt(sin (x+ pi//4)) + sqrt(7x-x^(2)-6)`
For f(x) to be real, we must have
`sin (x+(pi)/(4)) gt 0 and 7x - x^(2) - 6 gt 0`
`rArr 2npi le x +(pi)/(4) le (2n+1) pi and 1 le x le 6`, where `n in Z`
`rArr x in [1,(3pi)/(4)]uu[(7x)/(4) ,6]`