Q27 / 75 · JEE Main 2025 (4 Apr Shift 2) · Mathematics

Let f be a differentiable function on $\mathbf{R}$ such that $\mathrm{f}(2) = 1$, $f^{\prime}(2)=4$. Let $\lim _{x \rightarrow 0}(f(2+x))^{3 / x}=e^\alpha$. Then the number of times the curve $y=4 x^3-4 x^2-4(\alpha-7) x-\alpha$ meets x -axis is :-

  • Correct answer
    A
    $2$
  • B
    $1$
  • C
    $0$
  • D
    $3$

Solution

$\begin{aligned} & \lim _{x \rightarrow 0}(f(2+x))^{\frac{3}{x}} \\ & \lim _{\mathrm{e}^{x \rightarrow 0}} \frac{(f(2+x)-1) 3}{x} \\ & \mathrm{e}^{3 f^{\prime}(2)}=(e)^{12}=(e)^a \Rightarrow a=12 \\ & y=4 x^3-4 x^2-4(a-7) x-a \\ & y=4 x^3-4 x^2-20 x-12 \\ & \text { roots } x=-1,-1,3 \\ & \text { option }(1)\end{aligned}$

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

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