Q. 395.0( 1 Vote )

# Find the intervals in which f(x) is increasing or decreasing :i. f(x) = x |x|, x ϵ Rii. f(x) = sin x + |sin x|, 0 < x ≤ 2 πiii. f(x) = sin x (1 + cos x), 0 < x < π/2

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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