5
Rationalised 2023-24
APPLICATION OF DERIVATIVES
157
Also, the given function is continuous at x = 0 and 
6
x
=π Therefore, by Theorem 1,
f is increasing on 0 6
, π

  and decreasing on 
π π
6 2
,


 Example 13 Find the intervals in which the function f given by
 f (x) = sin x + cos x, 0 ≤ x ≤ 2π
is increasing or decreasing Solution We have
f(x) = sin x + cos x,
or
f ′(x) = cos x – sin x
Now 
( )
0
f
′x =
 gives sin x = cos x which gives  that 
4
x
=π
, 5
4
π  as 0
2
≤x
≤ π
The points 
4
x
=π
 and 
5
4
x
=π
 divide the interval [0, 2π] into three disjoint intervals,
namely, 0, 4
π



 , π
π
4
5
,4


  and 5 ,2
π4
π



