Application of DerivativeHard
Question
Let φ(x) = (f(x))3 - 3(f(x))2 + 4f(x) + 5x + 3 sin x + 4 cos x ∀ x ∈ R, then
Options
A.φ is increasing whenever f is increasing
B.φ is increasing whenever f is decreasing
C.φ is decreasing whenever f is decreasing
D.φ is decreasing if f′(x) = - 11
Solution
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 decreasing.
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 decreasing.
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