Q45 / 155 · WBJEE 2025 · Mathematics

If $x=-1$ and $x=2$ are extreme points of $f(x)=\alpha \log |x|+\beta x^2+x,(x \neq 0)$, then

  • A
    $\alpha=-6, \beta=\frac{1}{2}$
  • B
    $\alpha=-6, \beta=-\frac{1}{2}$
  • Correct answer
    C
    $\alpha=2, \beta=-\frac{1}{2}$
  • D
    $\alpha=2, \beta=\frac{1}{2}$

Solution

$f^{\prime}(x)=\frac{\alpha}{|x|} \cdot \frac{|x|}{x}+2 \beta x+1, f(-1)=f(2)=0$ So, $-\alpha-2 \beta+1=0 \Rightarrow \alpha+2 \beta-1=0$ ...(i) $\& \frac{\alpha}{2}+4 \beta+1=0 \Rightarrow \alpha+8 \beta+2=0$....(ii) (ii) - (i) $\Rightarrow 6 \beta+3=0 \Rightarrow \beta=-1 / 2$ (i) $\Rightarrow \alpha-1-1=0 \Rightarrow \alpha=2$

Hence, the correct answer is option: 'C'.

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