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Let $f : R \to R$ be a function such that $f (x) = ax + 3 \sin x + 4 \cos x$ .Then f(x) is invertible if

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a

a∈(−5,5)

b

a∈(−∞,5)

c

a∈(−5,+∞)

d

none of these

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detailed solution

Correct option is D

f′(x)=a+3cos ⁡x−4sin ⁡x =a+5cos ⁡(x+α), where cos ⁡α=35For invertible, f′(x)≥0∀x or f′(x)≤0∀x i.e.,a+5cos⁡(x+α)≥0 or a+5cos⁡(x+α)≤0 i.e.,a≥−5cos⁡(x+α) or a≤−5cos⁡(x+α) i.e.,a≥5 or a≤−5 f (x) must be monotonic. Thus,

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