Given function

$f(x)=x^3+5x+1, \qquad x\in \mathbb{R}$

We must check whether $f$ is:

• one-one (injective), and
• onto $\mathbb{R}$ (surjective).

Check if $f$ is one-one

A standard way is to study monotonicity using the derivative.

$f'(x)=3x^2+5$

Since

$3x^2+5>0 \quad \text{for every } x\in \mathbb{R},$

the function $f(x)$ is strictly increasing on all of $\mathbb{R}$.

A strictly increasing function cannot take the same value at two different points. Hence $f$ is one-one.

Check if $f$ is onto $\mathbb{R}$

Since $f$ is a cubic polynomial with positive leading coefficient, let us examine its end behavior:

• As $x\to +\infty$, $f(x)=x^3+5x+1\to +\infty$
• As $x\to -\infty$, $f(x)=x^3+5x+1\to -\infty$

Also, polynomial functions are continuous on $\mathbb{R}$.

Therefore, by continuity, $f(x)$ takes every real value between $-\infty$ and $+\infty$. Hence $f$ is onto $\mathbb{R}$.

Conclusion

The function is both:

• one-one, and
• onto $\mathbb{R}$.

So the correct option is:

$\boxed{\text{C: } f \text{ is one-one and onto } \mathbb{R}}$

Comparison with stored correct answer

Stored correct answer: C

Our derived answer: C

So they agree.