Question

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

• A. $\alpha =-6,\beta =\frac{1}{2}$
• B. $\alpha =-6,\beta =-\frac{1}{2}$
• C. $\alpha =2,\beta =-\frac{1}{2}$
• D. $\alpha =2,\beta =\frac{1}{2}$

Answer 1

The correct option is C $\alpha =2,\beta =-\frac{1}{2}$

$f^{\prime }\left(x\right)=\frac{\alpha }{x}+2\beta x+1$

$2\beta x^2+x+\alpha =0$ has roots $-1$ and $2$

Putting $x=-1$,

$2\beta -1+\alpha =0$

$2\beta +\alpha =1$

$8\beta +2+\alpha =0$

by solving above equations we get $\alpha =2,\beta =-\frac{1}{2}$

Ans is Option:$\left[C\right]$