If y=alog x+bx^2+x has extreme values at x=1 and x=2 , then

The correct answer is Option (3) → \( a=-\displaystyle\frac{2}{3}\ ,b=-\displaystyle\frac{1}{6} \)

At an extreme point, \( \frac{dy}{dx}=0 \).

    \( \displaystyle \frac{dy}{dx}=\frac{a}{x}+2bx+1 \)

At x = 1: \( a+2b+1=0 \).
At x = 2: \( \frac{a}{2}+4b+1=0 \), i.e. \( a+8b+2=0 \).

Subtracting the first equation from the second:

    \( \displaystyle 6b+1=0\;\Rightarrow\;b=-\frac{1}{6} \)

    \( \displaystyle a=-1-2b=-1+\frac{1}{3}=-\frac{2}{3} \)