QJEEM08Mae7df: JEE Main 2008 Mathematics - Class 12 JEEM08Mae7df Solution: Suppose the cubic has three distinct real roots where and .…

Question

Verified Solution • 2025–26 Session moderate

Suppose the cubic

{x^3} - px + q

has three distinct real roots
where

p&gt;0

and

q&gt;0

. Then which one of the following holds?

A The cubic has minima at $\sqrt {{p \over 3}}$ and maxima at $-\sqrt {{p \over 3}}$
B The cubic has minima at $-\sqrt {{p \over 3}}$ and maxima at $\sqrt {{p \over 3}}$
C The cubic has minima at both $\sqrt {{p \over 3}}$ and $-\sqrt {{p \over 3}}$
D The cubic has maxima at both $\sqrt {{p \over 3}}$ and $-\sqrt {{p \over 3}}$

Accepted Answer

Let

y = {x^3} - px + q

\Rightarrow {{dy} \over {dx}} = 3{x^2} - p

For

{{dy} \over {dx}} = 0 \Rightarrow 3{x^2} - p = 0

\Rightarrow x = \pm \sqrt {{p \over 3}}

{{{d^2}y} \over {d{x^2}}} = 6x

{\left. {{{{d^2}y} \over {d{x^2}}}} \right|_{x = \sqrt {{p \over 3}} }} = + ve\,\,\,\,

and

\,\,\,\,\,\,\,\,\,\,

{\left. {\,\,\,{{{d^2}y} \over {d{x^2}}}} \right|_{x = - \sqrt {{p \over 3}} }} = - ve

\therefore

y

has ninima at

x = \sqrt {{p \over 3}}

and maxima at

x = - \sqrt {{p \over 3}}

Step-by-Step Verified Solution