Let f be a differentiable function on R such that f(2) 1, f'(2) = 4. Let lim x→0 (f(2+x))^(3/x) = e^α.

Question

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

(1) 2

(2) 1

(3) 0

(4) 3  

Answer 1

Correct option is: (1) 2

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

\(e^{\lim _{x \rightarrow 0} \frac{3}{x}(f(2+x)-1)}=e^{\lim _{x \rightarrow} 3 f^{\prime}(2+x)} \)

\(=e^{3 f^{\prime}(2)} \)

\(=e^{12} \)

\(\Rightarrow \alpha=12 \)

\( y=4 x^3-4 x^2-4(12-7) x-12 \)

\( y=4 x^3-4 x^2-20 x-12 \)

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

\(=4(x+1)^2(x-3)\)  

It meets the x-axis at two points