 ## Book: RD Sharma - Mathematics (Volume 1)

### Chapter: 17. Increasing and Decreasing Functions

#### Subject: Maths - Class 12th

##### Q. No. 39 of Exercise 17.2

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39
##### 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

(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 , 1
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