Let $ f $ be a differentiable function on $ \mathbb{R} $ such that $ f(2) = 4 $. Let $ \lim_{x \to 0} \left( f(2+x) \right)^{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:
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For cubic equations, use numerical methods or approximation techniques to find the number of real roots.
We are provided with \( f(2) = 1 \) and \( f'(2) = 4 \). Let \( \alpha = \lim_{x \to 0^+} f(2 + x) \). Using a first-order Taylor expansion of \( f(2 + x) \) around \( x = 0 \):
\[f(2 + x) \approx f(2) + f'(2)x = 1 + 4x\]
Therefore, \( \alpha = 1 + 4x \). Substituting this into the equation of the curve:
\[y = 4x^3 - 4x^2 - 4(1 + 4x - 7)x - (1 + 4x)\]
Simplifying the equation:
\[y = 4x^3 - 4x^2 - 4(-6x) - 1 - 4x = 4x^3 - 4x^2 + 24x - 1 - 4x\]
\[y = 4x^3 - 4x^2 + 20x - 1\]
To determine how many times the curve intersects the x-axis, we set \( y = 0 \):
\[4x^3 - 4x^2 + 20x - 1 = 0\]
Numerical methods reveal that this cubic equation has two real roots. Consequently, the curve intersects the x-axis at two points. Thus, the answer is \( 2 \).
