Find the intervals in which f(x) is increasing or decreasing :
i. f(x) = x |x|, x ϵ R
ii. f(x) = sin x + |sin x|, 0 < x ≤ 2 π
iii. f(x) = sin x (1 + cos x), 0 < x < π/2
(i): Consider the given function,
f(x) = x |x|, x ϵ R
⇒ 
⇒ 
⇒ f
> 0
Therefore, f(x) is an increasing function for all real values.
(ii): Consider the given function,
f(x) = sin x +|sin x|, 0< x
2
⇒ 
⇒ 
The function 2cos x will be positive between (0,
)
Hence the function f(x) is increasing in the interval (0,
)
The function 2cos x will be negative between (
)
Hence the function f(x) is decreasing in the interval (
)
The value of f
= 0, when,
Therefore, the function f(x) is neither increasing nor decreasing in the interval (
)
(iii): consider the function,
f(x) = sin x(1 + cos x), 0 < x < 
⇒ f ’(x) = cos x + sin x( – sin x ) + cos x ( cos x )
⇒ f ’(x) = cos x – sin2 x + cos2 x
⇒ f ’(x) = cos x + (cos2 x – 1) + cos2 x
⇒ f ’(x) = cos x + 2 cos2 x – 1
⇒ f ’(x)=(2cos x – 1)(cos x + 1)
for f(x) to be increasing, we must have,
f’(x)> 0
⇒ f
)=(2
⇒ 0 < x< 
So, f(x) to be decreasing, we must have,
f
< 0
⇒ f
)=(2cos x – 1)(cos x + 1)
⇒ 
< x < 
⇒ x
,
So,f(x) is decreasing in 
,