Question

The values of $a$ and $b$ for which the function $y=a{\log }_{e}x+b{x}^2+x$, has extremum at the points $x_1=1$ and $x_2=2$ are :

• A. $a=\frac{-1}{3},b=\frac{-1}{6}$
• B. $a=\frac{2}{3},b=\frac{-1}{6}$
• C. $a=\frac{-2}{3},b=\frac{-1}{6}$
• D. $a=\frac{-2}{3},b=\frac{1}{6}$

Answer 1

The correct option is C $a=\frac{-2}{3},b=\frac{-1}{6}$

$\frac{dy}{dx}=\frac{a}{x}+2bx+1$
At extreme points,
$\frac{dy}{dx}=0$
$\Rightarrow \frac{dy}{dx}{|}_{x=1}=a+2b+1=0\cdots \left(1\right)$

and $\frac{dy}{dx}{|}_{x=2}=\frac{a}{2}+4b+1=0\dots \left(2\right)$

From $\left(1\right)$ and $\left(2\right)$,
$b=\frac{-1}{6},a=\frac{-2}{3}$