Let f be a differentiable function on R such that f(2)=1 and f'(2)=4. If lim x→0 (f(2+x))^(3/x)=e^(α), then the number of times the curve y=4x³-4x²-4(α-7)x-α meets the x -axis is:

Limit, Continuity and Differentiability: JEE Main 2025 Mathematics question

JEE Main 2025 · April, Unspecified (session-level dataset) · Q103

JEE Main 2025April, Unspecified (session-level dataset)MathematicsMediumLimit, Continuity and Differentiability420× since 2002

Let

f

be a differentiable function on

\mathbb{R}

such that

f(2)=1

and

f'(2)=4

. If

\displaystyle\lim_{x\to0}\bigl(f(2+x)\bigr)^{3/x}=e^{\alpha}

, then the number of times the curve

y=4x^{3}-4x^{2}-4(\alpha-7)x-\alpha

meets the

x

-axis is: