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

The cubic has minimum at $\sqrt{\frac{p}{3}}$ and maximum at $-\sqrt{\frac{p}{3}}$

Let $f(x)=x^3-p x+q$. Then,

$f'(x)=3 x^2-p=3\left(x-\sqrt{\frac{p}{3}}\right)\left(x+\sqrt{\frac{p}{3}}\right)$

The signs of f'(x) for different values of x are as shown below:

As f'(x) changes its sign from positive to negative in the neighbourhood of $-\sqrt{\frac{p}{3}}$.

So, $-\sqrt{\frac{p}{3}}$ is a point of local maximum. Similarly, $x=\sqrt{\frac{p}{3}}$ is a point of local minimum.