Question

Consider the function for $x=[-2,3]$, $f\left(x\right)=\{\begin{array}{cc}\frac{x^3-2x^2-5x+6}{x-1},& ifx\ne 1\\ -6& ifx=1\end{array}$ then

• A. $f$ is discontinuous at $x=1$ $\Rightarrow$ Rolles theorem is not applicable in $[-2,3]$
• B. $f(-2)\ne f\left(3\right)\Rightarrow$ Rolles theorem is not applicable in $[-2,3]$
• C. $f$ is not derivable in $(-2,3)\Rightarrow$ Rolles theorem is not applicable
• D. Rolles theorem is applicable as $f$ satisfies all the conditions and $c$ of Rolles theorem is $\frac{1}{2}$

Answer 1

The correct option is D Rolles theorem is applicable as $f$ satisfies all the conditions and $c$ of Rolles theorem is $\frac{1}{2}$

$f(-2)=f\left(3\right)=0$

$F\left(x\right)$ is continuous in $[-2,3]$ & derivable in $(-2,3)$ so Rolles theorem is applicable.

So $\exists cϵ(-2,3)$ such that $f‘\left(c\right)=0$

$\Rightarrow \frac{2c^3-5c^2+4c-1}{(c-1{)}^2}=0$

After solving this, we get

$\Rightarrow c=\frac{1}{2}$