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Suppose the cubic x^(3)-px+q has three d...

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

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))`

Text Solution

Verified by Experts

The correct Answer is:
C
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)`
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