If y = a log |x| + bx^2 + x has extreme values at x = -1 and at x = 2…

The correct answer is Option (3) → $a=2,b=−1/2$

Given $y = a \log |x| + bx^2+x$, clearly $D_f= R$ except 0.

Differentiating w.r.t. x, we get

$\frac{dy}{dx}=a.\frac{1}{x}+2bx+1$

For extreme points, $\frac{dy}{dx}=0$

$⇒\frac{a}{x}+2bx+1=0⇒a+2bx^2 + x = 0$.

Since $x = -1$ and $x = 2$ are extreme points,

$∴ a + 2b-1 = 0$ and $a + 8b + 2 = 0$.

Solving these equations, we get $a = 2, b =-\frac{1}{2}$