JEE Mains · Maths · STD 11 - 12. limits
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 :-
- A \(2\)
- B \(1\)
- C \(0\)
- D \(3\)
Answer & Solution
Correct Answer
(A) \(2\)
Step-by-step Solution
Detailed explanation
\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 }…
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