The condition f(x)=x3+p x2+q x+ r(x ∈ R) to have no extreme value, is

Question

The condition f(x)=x3+p x2+q x+ r(x ∈ R) to have no extreme value, is (A) p2< 3 q (B) 2 p2< q (C) p2< (1/4) q (D) p2 >3 q

Tardigrade Admin · Accepted Answer

Given, f(x)=x3+p x2+q x +r Now, f'(x)=3 x2+2 x p +q Clearly f'(x) >0 Now, b2-4 a c< 0 ⇒ 4 p2-4 × 3 × q< 0 ⇒ 4 p2-12 q< 0 ⇒ p2< 3 q

Q. The condition $f (x) = x^{3} + p x^{2} + q x + r (x \in R)$ to have no extreme value, is