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JEE Main 20254 AprilEvening Shift
Exam
JEE Main
Year
2025
Date
4 April
Shift
Evening Shift
Subject
Mathematics
Chapter
Application of Derivatives

JEE Main 2025 · 4 April · Evening Shift · Mathematics · Application of Derivatives

Let f be a differentiable function on R such that f(2) = 1 , f^...

Let f be a differentiable function on R\mathbf{R} such that f(2)=1\mathrm{f}(2) = 1, f′(2)=4f^{\prime}(2)=4. Let lim⁡x→0(f(2+x))3/x=eα\lim _{x \rightarrow 0}(f(2+x))^{3 / x}=e^\alpha. Then the number of times the curve y=4x3−4x2−4(α−7)x−αy=4 x^3-4 x^2-4(\alpha-7) x-\alpha meets x -axis is :-

Options

  1. A22
  2. B11
  3. C00
  4. D33

Correct answer

A. 22

Step-by-step solution

lim⁡x→0(f(2+x))3xlim⁡ex→0(f(2+x)−1)3xe3f′(2)=(e)12=(e)a⇒a=12y=4x3−4x2−4(a−7)x−ay=4x3−4x2−20x−12 roots x=−1,−1,3 option (1)\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}
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