The critical points of $f(x)=x^3+x^2+x+1$ are

do not exist

$f(x)=x^3+x^2+x+1$

differentiating wrt (x)

$\frac{d f(x)}{d x}=3 x^2+2 x+1=0$         ......(1)

solving eq (1) for roots to get critical points

$3 x^2+2 x+1=0$

for this quadratic equation

Discriminant = $\left(b^2-4 a c\right)$

$=(2)^2-4(1)(3)

$=4-12$

$=-9$

Discriminant < 0 ⇒ Imaginary roots exist

Option 4 ⇒ critical points don't exist