If f(x)=|x-2|, x in[0,4] then the Rolle's the | vedclass

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(A) For Rolle's Theorem to be applicable to a function $f(x)$ on $[a, b]$,three conditions must be satisfied: $1$. $f(x)$ is continuous on $[a, b]$. $

Vedclass · Accepted Answer

The function is not differentiable at every point in the $(0,4)$.

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(A) For Rolle's Theorem to be applicable to a function $f(x)$ on $[a, b]$,three conditions must be satisfied: $1$. $f(x)$ is continuous on $[a, b]$. $2$. $f(x)$ is differentiable on $(a, b)$. $3$. $f(a) = f(b)$. Given $f(x) = |x-2|$ on $[0, 4]$: Calculate $f(0) = |0-2| = |-2| = 2$. Calculate $f(4) = |4-2| = |2| = 2$. Wait,$f(0) = 2$ and $f(4) = 2$,so $f(0) = f(4)$. However,check differentiability: $f(x) = |x-2|$ is not differentiable at $x = 2$,which lies in the interval $(0, 4)$. Since the function is not differentiable at $x = 2$,Rolle's Theorem cannot be applied.

List-$I$	List-$II$
$A. \|x-1\|$	$I. 2 \log (e^3+e^2)$
$B. \log x$	$II. 2$
$C. x^2+x+1$	$III. \log_3 e^2$
$D. e^x$	$IV. \sqrt{2}$
	$V. \log \left(\frac{e^3-e}{2}\right)$

If $f(x)=|x-2|, x \in[0,4]$ then the Rolle's theorem cannot be applied to the function because

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