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

Answer

B

Explanation

For maxima or minima, put f' (x) = 0
Let `f(x)=x^(3)-px+q`
Then, `f'(x)=3x^(2)-p`
Put `f'(x)=0`
`implies" "x=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)lt0,` maxima