Let f be a differentiable function on $\mathbf{R}$ such that $\mathrm{f}(2)=1$, $f'(2)=4$. 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 :-

JEE Main · 2025 · 04 Apr 2025 · Shift 2 · Limits

Let f be a differentiable function on

R

such that

f (2) = 1

f^{'} (2) = 4

. Let

lim_{x \to 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 :-

Answer: (A)

Since

f (2) = 1

, the limit has the form

1^{\infty}

. Writing it as

exp (lim_{x \to 0} \frac{3 ( f ( 2 + x ) - 1 )}{x}) = exp (3 f^{'} (2)) = e^{12}

, so

α = 12

. The curve becomes

y = 4 x^{3} - 4 x^{2} - 20 x - 12 = 4 (x + 1)^{2} (x - 3)

, whose zeros are

x = - 1

(repeated) and

x = 3

. The graph therefore meets the

x

-axis at two distinct points.

JEE Main · 2025 · 04 Apr 2025 · Shift 2 · Single correct option

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