Answer :

(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 ,


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