Examine if Rolle's Theorem is applicable to any of the following functions. Can you say some thing about the converse of Roller's Theorem from these examples?
$f(x)=x^{2}-1$ for $x \in[1,2]$
Examine if Rolle's Theorem is applicable to any of the following functions. Can you say some thing about the converse of Roller's Theorem from these examples?
$f(x)=x^{2}-1$ for $x \in[1,2]$
By Rolle's Theorem, for a function $f:[a, b] \rightarrow R,$ if
a) $f$ is continuous on $[a, b]$
b) $f$ is continuous on $(a, b)$
c) $f(a)=f(b)$
Then, there exists some $c \in(a, b)$ such that $f^{\prime}(c)=0$
Therefore, Rolle's Theorem is not applicable to those functions that do not satisfy any of the three conditions of the hypothesis.
$f(x)=x^{2}-1$ for $x \in[1,2]$
It is evident that $f$, being a polynomial function, is continuous in $[1,2]$ and is differentiable in $(1,2).$
$f(1)=(1)^{2}-1=0$
$f(2)=(2)^{2}-1=3$
$\therefore f(1) \neq f(2)$
It is observed that $f$ does not satisfy a condition of the hypothesis of Roller's Theorem.
Hence, Roller's Theorem is not applicable for $f(x)=x^{2}-1$ for $x \in[1,2].$
Similar Questions
In $[0, 1]$ Lagrange's mean value theorem is $ NOT$ applicable to
In $[0, 1]$ Lagrange's mean value theorem is $ NOT$ applicable to
- [IIT 2003]
Which of the following function can satisfy Rolle's theorem ?
Which of the following function can satisfy Rolle's theorem ?
A value of $c$ for which the conclusion of mean value the theorem holds for the function $f(x) = log{_e}x$ on the interval $[1, 3]$ is
A value of $c$ for which the conclusion of mean value the theorem holds for the function $f(x) = log{_e}x$ on the interval $[1, 3]$ is
Let $f$ and $g$ be real valued functions defined on interval $(-1,1)$ such that $g^{\prime \prime}(x)$ is continuous, $g(0) \neq 0, g^{\prime}(0)=0, g^{\prime \prime}(0) \neq$ 0 , and $f(x)=g(x) \sin x$.
$STATEMENT$ $-1: \lim _{x \rightarrow 0}[g(x) \cot x-g(0) \operatorname{cosec} x]=f^{\prime \prime}(0)$.and
$STATEMENT$ $-2: f^{\prime}(0)=g(0)$.
Let $f$ and $g$ be real valued functions defined on interval $(-1,1)$ such that $g^{\prime \prime}(x)$ is continuous, $g(0) \neq 0, g^{\prime}(0)=0, g^{\prime \prime}(0) \neq$ 0 , and $f(x)=g(x) \sin x$.
$STATEMENT$ $-1: \lim _{x \rightarrow 0}[g(x) \cot x-g(0) \operatorname{cosec} x]=f^{\prime \prime}(0)$.and
$STATEMENT$ $-2: f^{\prime}(0)=g(0)$.
- [IIT 2008]
If $f$ is a differentiable function such that $f(2x + 1) = f(1 -2x)$ $\forall \,\,x \in R$ then minimum number of roots of the equation $f'(x) = 0$ in $x \in \left( { - 5,10} \right)$ ,given that $f(2) = f(5) = f(10)$ , is
If $f$ is a differentiable function such that $f(2x + 1) = f(1 -2x)$ $\forall \,\,x \in R$ then minimum number of roots of the equation $f'(x) = 0$ in $x \in \left( { - 5,10} \right)$ ,given that $f(2) = f(5) = f(10)$ , is