Question

Roll's theorem is applicable on following function. If this statement is true enter 1 otherwise 0.

$f\left(x\right)=x$ ${}^2$ $-1$ for $x$ $ϵ$ $[1,2]$

Answer 1

By Rolles Theorem, for a function $f:[a,b]\to R$,
if (a) $f$ is continuous on $[a,b]$
(b) $f$ is differentiable on $(a,b)$ and
(c) $f\left(a\right)=f\left(b\right)$
Then, there exists some $c\in (a,b)$ such that $f^{\prime }\left(c\right)=0$
Therefore, Rolles Theorem is not applicable to those functions that do not satisfy any of the three conditions of the hypothesis.
$f\left(x\right)=x^2-1$ for $x\in [1,2]$
It is evident that $f$, being a polynomial function, is continuous in $[1,2]$ and is differentiable in $(1,2).$
Also $f\left(1\right)=(1{)}^2-1=0$
and $f\left(2\right)=(2{)}^2-1=3$
$\therefore f\left(1\right)\ne f\left(2\right)$
It is observed that $f$ does not satisfy a condition of the hypothesis of Rolles Theorem.
Hence, Rolles Theorem is not applicable for $f\left(x\right)=x^2-1$ for $x\in [1,2].$