The condition f(x)=x^3+p x^2+q x+r(x ∈ R) to have no extreme value, is

If $f(x)=x^3+p x^2+q x+r$ has no extreme values, then

$f'(x) \neq 0$ for any $x \in R$

$\Rightarrow 3 x^2+2 p x+q \neq 0$ for any $x \in R$

$\Rightarrow 3 x^2+2 p x+q=0$ has no real root

$\Rightarrow 4 p^2-12 q<0 \Rightarrow p^2<3 q$.