Question

Suppose the cubic $x^3-px+q$ has three distinct real roots where $p>0$ and $q>0$. Then which one of the following holds?

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

Answer 1

The correct option is C The cubic has minima at $\sqrt{\frac{p}{3}}$ and maxima at $-\sqrt{\frac{p}{3}}$

Let $f\left(x\right)=x^3-px+q$
$\Rightarrow f^{\prime }\left(x\right)=3x^2-p$
For maxima or minima
$f^{\prime }\left(x\right)=0\Rightarrow x=\pm \sqrt{\frac{p}{3}}$
$f^{\prime \prime }\left(x\right)=6x\Rightarrow f^{\prime \prime }\left(x\right)>0$ for $x=\sqrt{\frac{p}{3}}$
and $f^{\prime \prime }\left(x\right)<0$ for $x=-\sqrt{\frac{p}{3}}$