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Question

Question: If f : R → R where f(x) = ax + cosx is an invertible function then...

If f : R → R where f(x) = ax + cosx is an invertible function then

A

ae(-2, 1]∪[1, 2)

B

a ∈[-2, 2]

C

ae[-∞, -1]∪[1, ∞)

D

a ∈ [-1, 1]

Answer

ae,1-∞, -1∪[1, ∞)

Explanation

Solution

The function f(x) is clearly on to if a ≠ 0. Now f '(x)

= a − sinx. f '(x) > 0 ∀ x ∈ R

⇒ a ≥ sin x ∀ x ∈ R

⇒ a ≥ 1. For f'(x) ≤ 0. ∀ x ∈ R

we get a ≤ sin x ∀ x ∈ R

⇒ a ≤ -1 a ∈ (-∞, -1]∪[1, ∞)