Question

If $f\left(x\right)=x^3+3x^2+12x-2\sin x$ such that $f:R\to R$ then

• A. $f\left(x\right)$ is one-one and onto
• B. $f\left(x\right)$ is many one onto
• C. $f\left(x\right)$ is one-one and into
• D. $f\left(x\right)$ is many one into

Answer 1

The correct option is A $f\left(x\right)$ is one-one and onto

We have,
$f\left(x\right)=x^3+3x^2+12x-2\sin x$
Since $-1\le \sin x\le 1$ and $-\infty <x^3+3x^2+12x<\infty$
Range of f is $R$ which is equal to co-domain $\Rightarrow$f is onto function.
Now $f^{\prime }\left(x\right)=3x^2+6x+12-2\cos x$
Also $-1\le \cos x\le 1$
$\Rightarrow f^{\prime }\left(x\right)=3x^2+6x+12-2\left(1\right)=3x^2+6x+10$
Discriminant of above quadratic is $36-4\left(3\right)\left(10\right)<0$
which means $f^{\prime }\left(x\right)>0\forall x\in R\Rightarrow f$ is increasing function $\Rightarrow$ f is an one-one function
Hence option 'A' is correct choice.