Question

Let $a,b\in R$ be such that the function $f$ given by $f\left(x\right)=\ln |x|+b{x}^2+ax,x\ne 0$ has extreme values at $x=-1$ and $x=2$.
Statement 1 : $f$ has local maximum at $x=-1$ and at $x=2$
Statement 2 : $a=\frac{1}{2}$, $b=\frac{-1}{4}$

• A. Statement 1 is false, Statement 2 is true
• B. Statement 1 is true, Statement 2 is true; Statement 2 is correct explanation for Statement 1
• C. Statement 1 is true, Statement 2 is true; Statement 2 is not a correct expalnation for Statement 1.
• D. Statement 1 is true, Statement 2 is false.

Answer 1

The correct option is B Statement 1 is true, Statement 2 is true; Statement 2 is correct explanation for Statement 1

$f^{\prime }\left(x\right)=\frac{1}{x}+2bx+a$
$f$ has extreme values and is differentiable.
$\Rightarrow f^{\prime }(-1)=0\Rightarrow a-2b=1$
$\Rightarrow f^{\prime }(-2)=0\Rightarrow a+4b=\frac{-1}{2}\Rightarrow a=\frac{1}{2},b=\frac{-1}{4}$
$f^{\prime \prime }(-1)$ and $f^{\prime \prime }(-2)$ are negative .Therefore, $f$ has local maxima at $-1$ and $2$
Hence, option 'B' is correct.