Question

Consider the function for $x=[−2,3],f(x)=\{\begin{array}{llc}\frac{x^3−2x^2−5x+6}{x−1}& \text{ if }& xî€=1\\ −6& \text{ if }& x=1\end{array}$, then

• A. $f$ is discontinuous at $x=1⇒$ Rolle's theorem is not applicable in $[−2,3]$
• B. $f(−2)î€=f(3)⇒$ Rolle's theorem is not applicable in $[−2,3]$
• C. $f$ is not derivable in $(−2,3)⇒$ 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 $∃c∈(−2,3)$ such that $f^{′}(c)=0$
$⇒\frac{2c^3−5c^2+4c−1}{(c−1{)}^2}=0⇒c=1/2$