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

Options

A. The cubic has maxima at both `sqrt(p/3)` and `-sqrt(p/3)`
B. The cubic has minima at `sqrt(p/3)` and maxima at `-sqrt(p/3)`
C. The cubic has minima st `-sqrt(p/3)` and maxima at `sqrt(p/3)`
D. The cubic has minima at both `sqrt(p/3)` and `-sqrt(p/3)`

Answer

B

Explanation

Let `f(x)=x^(3)=px+q`
Then `f'(x)=3x^(2)-p`
Put `f'(x)=0impliesx=sqrt(p/3)-sqrt(p/3)`
Now `f''(x)=6x`
`:.` at `x=sqrt(p/3),f''(x)=6sqrt(p/3) gt0` (Minima)
and at `x=-sqrt(p/3), f''(x)=-6 sqrt(p/3)lt0` (Maxima0