Let  $f\left(x\right)=x^4-4x^3+12x^2+x-1.$ Then which of the following statements are True
 $S_1: f\left(x\right)=0$  has two real and distinct roots 
$S_2 :f^1\left(x\right)=0$ has one real root

 $S_3 :f^{11}\left(x\right)=0$ has two real and distinct roots

Solution:  $f^1\left(x\right)=4x^3-12x^2+24x+1$ and  $f^{11}\left(x\right)=12x^2-24x+24=12[{(x-1)}^2+1]>0$
$f^1\left(x\right)$ is increasing only. So  $f^1\left(x\right)=0,3^{rd}$ degree polynomial equation, has only one root
 $f \left(x\right)$ is concave up only and $f(-1)>0,f\left(0\right)<0,f\left(1\right)>0$