Let φ (x) = (f(x)) 3 – 3(f(x)) 2 + 4f(x) + 5x + 3 sin x + 4 cos x ∀ x ∈ R, where f(x) is a differentiable function ∀ x ∈ R, then
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(a,d) φ′ (x) = (3(f(x)) 2 – 6(f(x)) + 4)f ′ (x) + 5 + 3 cos x – 4 sin x
5 – 
≤ 5 + 3 cosx – 4 sin x ≤ 5 + 
adding (3(f(x)) 2 – 6(f(x)) + 4)f ′ (x)
(3(f(x)) 2 – 6(f(x)) + 4)f ′ (x) ≤ φ′ (x) ≤ (3(f(x)) 2 – 6(f(x)) + 4)f ′ (x) + 10
3(f(x)) 2 – 6f(x) + 4 = 3 (f(x) – 1) 2 + 1 > 0
(3(f(x)) 2 – 6(f(x)) + 4)f ′ (x) ≥ 0 when ever f(x) is increasing.
⇒ φ′ (x) ≥ 0 ⇒ φ (x) is increasing, when ever f(x) is increasing.
If f ′ (x) = – 11 then
(3(f(x)) 2 – 6f(x) + 4) f ′ (x) + 10 = – 33 (f(x) – 1) 2 – 1 < 0
⇒ φ′ (x) < 0
⇒ φ (x) is decreasing.
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