Question

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

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

Answer 1

Answer: (B)

$f^{\prime }\left(x\right)=\frac{1}{x}+2bx+a$
at $x=-1-1-2b+a=0$
$a-2b=1...\left(i\right)$
at $x=2\frac{1}{2}+4b+a=0$
$a+4b=-\frac{1}{2}...\left(ii\right)$
On solving $\left(i\right)$ and $\left(ii\right)a=\frac{1}{2},b=\frac{1}{4}$
$f^{\prime }\left(x\right)=\frac{1}{x}-\frac{x}{2}+\frac{1}{2}=\frac{2-x^2+x}{2x}=\frac{-\left(x+1\right)\left(x-2\right)}{2x}$
So maxima at $x=-1,2$