Answer
No critical numbers
Work Step by Step
Find the critical numbers for $f(x) = 2x^3+x^2+2x$
Differentiate and set the derivative $=0$
$f'(x) = 6x^2+2x+2$
$0=2(3x^2+x+1)$
$b^2-4ac = 1-4(3)(1) = -11$
The discriminant of the right hand side quadratic is $<0$. Therefore, the right hand side will never $=0$, so $f(x) $ has no critical numbers.
