First slide

Monotonicity

Question

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

Easy

A

$a \in (- 5, 5)$
B

$a \in (- \infty, 5)$
C

$a \in (- 5, + \infty)$
D

none of these

Solution

$f^{'} (x) = a + 3 \cos x - 4 \sin x = a + 5 \cos (x + α), where \cos α = \frac{3}{5}$
For invertible,
$f^{'} (x) \geq 0 \forall x or f^{'} (x) \leq 0 \forall x i . e ., a + 5 \cos (x + α) \geq 0 or a + 5 \cos (x + α) \leq 0 i . e ., a \geq - 5 \cos (x + α) or a \leq - 5 \cos (x + α) i . e ., a \geq 5 or a \leq - 5$ f (x) must be monotonic. Thus,

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