Question

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

• A. $p^{2}< 3 q$
• B. $2 p^{2}< q$
• C. $p^{2}< \frac{1}{4} q$
• D. $p^{2} >3 q$

Answer 1

Answer: (A)

Given, $f(x)=x^{3}+p x^{2}+q x +r$
Now, $f'(x)=3 x^{2}+2 x p +q$
Clearly $f'(x) >0$
Now, $b^{2}-4 a c< 0$
$\Rightarrow 4 p^{2}-4 \times 3 \times q< 0$
$\Rightarrow 4 p^{2}-12 q< 0$
$\Rightarrow p^{2}< 3 q$