Question

Consider the function for $x=[-2,3], f(x)=\begin{cases}\frac{x^3-2 x^2-5 x+6}{x-1} & \text { if } & x \neq 1 \\ -6 & \text { if } & x=1\end{cases}$, then

• A. $f$ is discontinuous at $x =1 \Rightarrow$ Rolle's theorem is not applicable in $[-2,3]$
• B. $f (-2) \neq f (3) \Rightarrow$ Rolle's theorem is not applicable in $[-2,3]$
• C. $f$ is not derivable in $(-2,3) \Rightarrow$ Rolle's theorem is not applicable
• D. Rolle's theorem is applicable as $f$ satisfies all the conditions and c of Rolle's theorem is $1 / 2$

Answer 1

Answer: (D)

$f (-2)= f (3)=0$
$f ( x )$ is continuous in $[-2,3]$ \& derivable in $(-2,3)$ so Rolle's theorem is applicable.
so $\exists c \in(-2,3)$ such that $f^{\prime}(c)=0$
$\Rightarrow \frac{2 c^3-5 c^2+4 c-1}{(c-1)^2}=0 \Rightarrow c=1 / 2$