If x = -1 and x = 2 are extreme points of f(x) = `alpha log|x| + beta x^2 + x`, then

Options

A. `alpha=2, beta=1/2`
B. `alpha=2,beta=1/2`
C. `alpha=-6,beta=1/2`
D. `alpha=-6,beta=-1/2`

Answer

A

Explanation

Given `f(x)=alpha In |x|+betax^(2)+x`………….i
`impliesf'(x)=(alpha)/x+2beta x+1=(2betax^(2)+x +alpha)/x`
Since `x=-1,2` are extreme points,
`:.f'(x)=0` at these points.
Hence `2beta-1+alpha=0`………………….ii
and `8 beta+2 +alpha=0` ………………….ii
On solving eqs ii and iii we get
`beta=(-1)/2, alpha=2`