Q. The condition $f(x) = x^{3} + px^{2} + qx + r(x \in R)$ to have no extreme value, is
A) $p^{2} < 3q$
B) $2p^{2} < q$
C) $p^{2} < \frac{1}{4}q$
D) $p^{2} > 3q$
Explanation
(a)
Given, $f(x) = x^{3} + px^{2} + qx + r$
$\Rightarrow \ \ \ f^{'}(x) = 3x^{2} + 2xp + q$ or $f^{'}(x) < 0$
Clearly, $f^{'}(x) > 0$
Now, $b^{2} - 4ac < 0$
$\Rightarrow \ \ \ \ \ \ 4p^{2} - 4 \times 3 \times q < 0$
$\Rightarrow \ \ \ \ \ p^{2} < 3q$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.