Suppose the cubic `x^(3)-px+q` has three distinct real roots, where `pgt0` and `qgt0`. Then find the max and min points .

Options

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

Answer

C

Explanation

Let `f(x)=x^(3)-px+q`
`:.f'(x)=3x^(2)-p`
`impliesf''(x)=6x`

For maxima or minima `f'(x)=0`
`:.x=+-sqrt(p/3)`
`impliesf''(sqrt(p/3))=6sqrt((p/3))gt0`
and `f''(-sqrt(p/3)=-6sqrt(p/3)lt0`
Hence given cubic minima at `x=sqrt(p/3)` and maxima at `x=-sqrt(p/3)`