Find the values of a for which the function f(x) = cos x -sin x + ax + b is decreasing.

f(x) = cos x - sin x + ax + b
df/dx = -sin x -cos x + a
For function to be decreasing, df/dx<0
Hence, -sinx - cosx +a <0
sinx + cosx>a
or root(2) sin (x+pi/4) > a
sin (x + pi/4) > a/root2
a/root2< -1
a< -root2
Hence, the value of a should be (-inf, -root 2) for the function to be decreasing.

Thanks & Regards
Bharat Bajaj
askIITians Faculty
IIT Delhi