500 followers problem!!!

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

$g(x)=f'(x)=9x^2+4x+1$

Now $g(x)$ is always positive as coefficient of $x^2>0$ and $∆<0$

Therefore, $f(x)$ is strictly increasing and hence $n=1$ .

• $f(0)=5>0$
• $f(-1)=3>0$
• $f(-2)=-13<0$

Thus $f(x)=0$ has a solution between $x=-2$ and $x=-1$ .

$\Rightarrow \lfloor l\rfloor=-2$

$n-\lfloor l \rfloor=1+2=3$

A plot of the curve clearly shows that $n =1$ and $\lfloor l \rfloor = -2$ . Therefore, $n - \lfloor l \rfloor = \boxed{3}$ .