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?
Answer & explanation
Correct answer: option 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.