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

f'(x)=-sinx-cox+a

Check the monotonicity of f(x) = x sin x in (0; =2)

Approved

If f(x) = R x2 1=x2 sin( p t)dt, then nd the value of f0(1).

Let a = min  x2 + 4x + 5; x 2 R and b = lim!0 1??cos 2 2 then nd the value of a??b a+b .

f(x) = cos x - sin x + ax + b

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

Approved

f'(x)=-sinx-cosx+a <0
the minmum value of -sinx-cosx, occurs at pi/4 = -sqrt(2)=-1.41
a-1.41<0
a<1.41

Sher Mohammad
B.Tech, IIT Delhi