Suppose the cubic x3−px+q=0  has three distinct real roots where p>0  and q>0 . Then which one of the following holds?

The given equation is f(x)=x3−px+q ⇒ f'(x)=3x2−p ⇒ f''(x)=6x for max. or min. f'(x)=0 ⇒3x2−p=0  ⇒x=±p3 Then f''(p3)=6(p3)>0 and  f''(−p3)=−6p3<0 hence, given cubic minima at x=P3  and maxima at x=−P3