Let $f$ be a differentiable function on $\mathbf{R}$ such that $f(2)=1, f^{\prime}(2)=4$. Let $\lim \limits_{x \rightarrow 0}(f(2+x))^{3 / x}=\mathrm{e}^\alpha$. Then the number of times the curve $y=4 x^3-4 x^2-4(\alpha-7) x-\alpha$ meets $x$-axis is :
Solution
<p>$$\begin{aligned}
&\begin{aligned}
& \lim _{x \rightarrow 0}(f(2+x))^{3 / x}=\left(1^{\infty} \text { form }\right) \\
& e^{\lim _{x \rightarrow 0 x}^3(f(2+x)-1)}=e^{\lim _{x \rightarrow} 3 f^{\prime}(2+x)} \\
& =e^{3 f^{\prime}(2)} \\
& =e^{12} \\
& \Rightarrow \alpha=12 \\
& y=4 x^3-4 x^2-4(12-7) x-12 \\
& y=4 x^3-4 x^2-20 x-12 \\
& y=4\left(x^3-x^2-5 x-3\right) \\
& =4(x+1)^2(x-3)
\end{aligned}\\
&\text { It meets the } x \text {-axis at two points }
\end{aligned}$$</p>
About this question
Subject: Mathematics · Chapter: Limits, Continuity and Differentiability · Topic: Limits and Standard Results
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