The condition that f ( x ) = a x 3 + b x 2 + c x + d has no extreme value, is

Given curve is $f\left(x\right)=a{x}^3+b{x}^2+cx+d$

On differentiating w.r.t. $x,$ we get

$f^{\text{'}}\left(x\right)=3a{x}^2+2bx+c$

For extremum, $f^{\text{'}}\left(x\right)=0\Rightarrow 3a{x}^2+2bx+c=0$

Since, it has no extremum value.

$\therefore \text{Discriminant}<0$

$\Rightarrow (2b{)}^2-4\times 3a\times c<0$

$\Rightarrow 4b^2-12ac<0\Rightarrow b^2-3ac<0$

$\Rightarrow b^2<3ac$