Suppose the cubic `x^3-px+q` has three real roots where `pgt0` and `qgt0` . Then which one of the following holds ?

Options

A. The cubic has minimum at both `sqrt(p/3)` and -`sqrt(p/3)`
B. The cubic has maximum at both `sqrt(p/3)` and -`sqrt(p/3)`
C. The cubic has minimum at `sqrt(p/3)` and maximum at `- sqrt(p/3)`
D. The cubic has minimum at `-sqrt(p/3)` and maximum at `sqrt(p/3)`

Answer

C

Explanation

Let `f(x)=x^3-px+q` Then
`f'(x)=3x^2-p=3(x-sqrt(p/3))(x+sqrt(p/3))`
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 neighourhood of `-sqrt(p/3)`.
So `-sqrt(p/3)` is a point of a local maximum . Similarly `x=sqrt(p/3)` is a point of local minimum.