The graph of y=x^3+a x^2+b x+c has no extremum if and only if

We have,

$y=x^3+a x^2+b x+c \Rightarrow \frac{d y}{d x}=3 x^2+2 a x+b$

Clearly, $\frac{d y}{d x}>0$ if $4 a^2-12 b<0$

$\Rightarrow y=f(x)$ will increase continuously if $a^2<3 b$

Hence, $f(x)$ has no extremum iff $a^2<3 b$