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

$\alpha=-6,\beta=\frac{1}{2}$

$\alpha=-6,\beta=-\frac{1}{2}$

$\alpha=2,\beta=-\frac{1}{2}$

$\alpha=2,\beta=\frac{1}{2}$

Option C

Here , x=-1 and x=2 are extreme points of $f(x)=\alpha\log|x|+\beta x^{2}+x$ , then

$f '(x)=\frac{\alpha}{x}+2\beta x+1$

$f '(-1)=-\alpha-2\beta+1=0$ ........(i)

[ At extreme point, f '(x0=0]

$f'(2)=\frac{\alpha}{2}+4\beta+1=0$ .......(ii)

On solving Eqs. (i) and (ii) , we get

$\alpha=2,\beta=-1/2$

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