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. $a\in (-5,5)$
• B. $a\in (-\infty ,5)$
• C. $a\in (-5,\infty )$
• D. None of the above

Answer 1

The correct option is D None of the above

Given : $f\left(x\right)=ax+3\sin x+4\cos x$
$\to$As $\sin x$ and $\cos x$ are periodic function of period : $2\pi$

$\to$ The function ${}^{\prime }{f}^{\prime }$ is invertible provided that the domain is restricted to an interval of length $2\pi$ or less than that.

Hence the option given $(-5,5),(-\infty ,5)and(-5,\infty )$ does not fall under the interval $2\pi$

Hence none of the above is the correct answer.

Therefore option D is correct Answer