Find the intervals in which the function f given by f (x) = (4sin x - 2x - x cos x)/(2 + cos x) is
(i) Increasing   (ii) Decreasing

Solution:

According to the given problem

We have

f (x) = (4sin x - 2x - x cos x)/(2 + cos x)

On differentiating wrt x, we get

f' (x) = [(2 + cos x)(4 cos x - 2 - cos x + x sin x) - (4sin x - 2x - x cos x)(-sin x)]/[(2 + cos x)²]

= [6 cos x - 4 + 2x sin x + 3cos² x - 2 cos x + x sin x cos x + 4 sin² x - 2x sin x - x sin x cos x] / [(2 + cos x)²]

= [4 cos x - 4 + 3cos² x + 4 sin² x] / [(2 + cos x)²]

= [4 cos x - 4 + 3cos² x + 4 - 4 cos² x] / [(2 + cos x)²]

= [4 cos x - cos² x]/[(2 + cos x)²]

f' (x) = (cos x (4 - cos x)) / (2 + cos x)²

Now,

f' (x) = 0

⇒ cos x = 0 or cos x = 4

But cos x ≠ 4

Hence,

cos x = 0

x = π / 2, 3π / 2

Now, x = π / 2 and x = 3π / 2 divide (0, 2π) into three disjoint intervals

i.e., (0, π/2), (π / 2, 3π. /2) and (3π/2, 2π)

In intervals, (0, π / 2) and (3π/2, 2π),

f' (x) > 0

Thus, f (x) is increasing for 0 < x < x/2 and 3x/2 < x < 2π

In the interval (π/2, 3π/2) f' (x) < 0

Thus, f (x) is decreasing for π/2 < x < 3π/2

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NCERT Solutions Class 12 Maths - Chapter 6 Exercise ME Question 6

Find the intervals in which the function f given by f (x) = (4sin x - 2x - x cos x)/(2 + cos x) is (i) Increasing (ii) Decreasing

Summary:

The  intervals in which the function f given by f (x) = (4sin x - 2x - x cos x)/(2 + cos x) is increasing for 0 < x < x/2 and 3x/2 < x < 2π and decreasing for π/2 < x < 3π/2