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

Options

A. `[1,6]`
B. `[1,(3pi)/(4)] cup [(7pi)/(4),6]`
C. `[1, pi] cup [(7pi)/(4),6]`
D. None of these

Answer

B

Explanation

We have `f(x)=sqrt(sinx+cosx)+sqrt(7x-x^(2)-6)`
`=sqrt(sqrt(2)sin((pi)/(4)+x))+sqrt(7x-x^(2)-6)`
`f(x)` is defined if
(i) `7x-x^(2)-6 ge 0`
`implies 1 le x le 6 " (1)" `
(ii) ` sin((pi)/(4)+x) ge 0`
` x +(pi)/(4) in ...[-2pi,-pi] cup [0, pi] cup [2pi, 3pi]... `
`implies x in ...[-(pi)/(4),(3pi)/(4)] cup [(7pi)/(4),(11pi)/(4)]..." (2)" `
From (1) and (2), we get
`x in [1,(3pi)/(4)] cup [(7pi)/(4),6]`