Question
Let $$f:R \to R$$ be a function such that $$f\left( x \right) = ax + 3\sin \,x + 4\cos \,x.$$ Then $$f\left( x \right)$$ is invertible if :
A.
$$a\, \in \,\left( { - 5,\,5} \right)$$
B.
$$a\, \in \,\left( { - \infty ,\, - 5} \right)$$
C.
$$a\, \in \,\left( {5,\, + \infty } \right)$$
D.
none of these
Answer :
$$a\, \in \,\left( { - 5,\,5} \right)$$
Solution :
$$\eqalign{
& f'\left( x \right) = a + 3\cos \,x - 4\sin \,x \cr
& = a + 5\cos \left( {x + \alpha } \right),{\text{ where }}\cos \,\alpha = \frac{3}{5} \cr
& \therefore a - 5 \leqslant f'\left( x \right) \leqslant a + 5 \cr} $$
$$\therefore f'\left( x \right) > 0{\text{ if }}a + 5 > 0{\text{ i}}{\text{.e}}{\text{., }}a > - 5$$ and $$f'\left( x \right) < 0{\text{ if }}a - 5 <0{\text{ i}}{\text{.e}}{\text{., }}a < 5$$
Hence, $$f\left( x \right)$$ is strictly monotonic if $$a\, \in \,\left( { - 5,\,5} \right)$$ and hence it will be invertible.