The set of all real values of x for which the funciton `f(x) = sqrt(sin x + cos x )+sqrt(7x -x^(2) - 6)` takes real values is

Answer

3

Explanation

`f(x)=sqrt(sinx+cosx)+sqrt(7x-x^(2)-6)`
`=sqrt(sqrt(2)sin(x+(pi)/(4)))+sqrt((x-6)(1-x))`
Now, `f(x)` is defined if `sin(x+(pi)/(4)) ge 0 and (x-6)(1-x) ge 0`
i.e., ` 0 le x+(pi)/(4) le pi or 2pi le x+(pi)/(4) le 3pi and 1 le x le 6`
i.e., `-(pi)/(4) le x le (3pi)/(4) or (7pi)/(4) le x le (11pi)/(4) and 1 le x le 6`
`or x in [1,(3pi)/(4)]cup [(7pi)/(4),6]`
Integral values of x are `x=1,2, and 6`.