Answer:
Option C
Explanation:
$f(x)=x^{4}-4x^{3}+12x^{2}+x-1$
$f'(x)= 4x^{3}-12x^{2}+24x+1$
$f''(x)=12x^{2}-24x+24$
=$12(x^{2}-2x+2)$
=$12(x-1)^{2}+1)>0$, for all x
$\Rightarrow$ f'(x) is increasing
Since , f'(x) is cubic and increasing
$\Rightarrow$ f'(x) has only one real root and two imaginary roots
$\therefore$ f(x) cannot have all distinct root
$\Rightarrow$ Atmost 2 real roots
Now, f(-1)=15
f(0)=-1 and f(1)=9
$\therefore$ f(x) must have one root in (-1,0) and other in (0,1)
$\Rightarrow$ 2 real roots
