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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 ,