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प्रश्न
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.
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उत्तर
Here,
f(x) = `(4 sin x - 2x - x cos x)/(2 + cos x)`
`= (4 sin x)/(2 + cos x) - x`
∴ f(x) = `((2 + cos x)4 cos x - 4 sin x (- sin x))/(2 + cos x)^2 - 1`
`= (8 cos x + 4 cos^2 x + 4 sin^2 x)/(2 + cos x)^2 - 1`
`= (8 cos x + 4 - (2 + cos x)^2)/(2 + cos x)`
`= (4 cos x - cos^2 x)/((2 + cos x)^2)`
`= (cos x (4 - cos x))/(2 + cos x)^2`
because – 1 ≤ cos x ≤ 1
⇒ 4 - cos x > 0 and (2 + cos x)2 > 0
∴ f(x) > 0 or < 0 such that cos x > 0 or cos x < 0 respectively
∴ f(x) is increasing when 0 < x < `pi/2, (3pi)/2 < x < 2 pi`
And f(x) is decreasing when `pi/2 < pi < (3pi)/2`.
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