Let f be a differentiable function on R such that f(2)=1,f^'(2)=4.

Question

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

Accepted Answer

$$\begin{aligned}
&\begin{aligned}
& \lim _{x \rightarrow 0}(f(2+x))^{3 / x}=\left(1^{\infty} \text { form }\right) \
& e^{\lim _{x \rightarrow 0 x}^3(f(2+x)-1)}=e^{\lim _{x \rightarrow} 3 f^{\prime}(2+x)} \
& =e^{3 f^{\prime}(2)} \
& =e^{12} \
& \Rightarrow \alpha=12 \
& y=4 x^3-4 x^2-4(12-7) x-12 \
& y=4 x^3-4 x^2-20 x-12 \
& y=4\left(x^3-x^2-5 x-3\right) \
& =4(x+1)^2(x-3)
\end{aligned}\
\&\text { It meets the } x \text {-axis at two points }
\end{aligned}$$

Solutions · 1