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

Mathematics · Limits, Continuity and Differentiability

JEE Main 2025 — 4 April, Evening Shift — Question 18

Let $f$ be a differentiable function on $\mathbf{R}$ such that $f(2)=$ 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:

$\lim _{x \rightarrow 0}(f(2+x))^{3 / x}=\left(1^{\infty}\right.$ form $)$

\begin{aligned} & e^{\lim _{x \rightarrow 0} \frac{3}{x}(f(2+x)-1)}=e^{\lim _{x \rightarrow} 3 f^{\prime}(2+x)} \\& =e^{3 f^{\prime}(2)}\\& =e^{12} \\& \Rightarrow \alpha=12 \\& y=4 x^{3}-4 x^{2}-4(12-7) x-12 \end{aligned}

$y=4 x^{3}-4 x^{2}-20 x-12$

$y=4\left(x^{3}-x^{2}-5 x-3\right)$

$=4(x+1)^{2}(x-3)$

It meets the $x$ -axis at two points

Answer key and solution verified before publishing.

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