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. $a\in [4,\infty )$
• D. $a\in (-\infty ,-5]\cup [5,\infty )$

Answer 1

The correct option is D $a\in (-\infty ,-5]\cup [5,\infty )$

$f\left(x\right)=ax+3\sin x+4\cos x$
$f^{\prime }\left(x\right)=a+3\cos x-4\sin x$
$=a+5\cos (x+\alpha ),\text{where}\cos \alpha =\frac{3}{5}$
For invertible function, $f\left(x\right)$ must be monotonic and onto
$\Rightarrow f^{\prime }\left(x\right)\ge 0\forall x\text{or}{f}^{\prime }\left(x\right)\le 0\forall x$
$\Rightarrow a+5\cos (x+\alpha )\ge 0\text{or}a+5\cos (x+\alpha )\le 0$
$\Rightarrow a\ge -5\cos (x+\alpha )\text{or}a\le -5\cos (x+\alpha )$
$\Rightarrow a\ge 5$ or $a\le -5$
$\therefore a\in (-\infty ,-5]\cup [5,\infty )$
Also $f\left(x\right)$ is onto for $x\ne 0.$